Regularity of minima: An invitation to the dark side of the calculus of variations

Abstract

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic,
and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

Full text

to appear in “Applications of Mathematics”.
REGULARITY OF MINIMA: AN INVITATION TO THE DARK
SIDE OF THE CALCULUS OF VARIATIONS
GIUSEPPE MINGIONE
Abstract. I am presenting a survey of regularity results for both minima
of variational integrals, and solutions to non-linear elliptic, and sometimes
parabolic, systems of partial differential equations. I will try to take the reader
to the Dark Side...
Contents
1. Prologue: The Dark Side (came to Paseky. . . ) 1
2. The scalar case, and the phantom irregularity 2
3. The revenge of irregularity: the vectorial case 9
4. A new hope: partial regularity 11
5. Irregularity strikes back 26
6. The return of regularity: (p, q)-growth conditions 28
7. Variable growth exponents 35
References 40
1. Prologue: The Dark Side (came to Paseky. . . )
These are the “generalized” lecture notes of a course I gave at the Paseky school
of Mathematical Theory in Fluid Mechanics, at the end of June 2005; “generalized”
because they largely extend the presentation I offered at Paseky. The school has
a great and prestigious tradition: it was founded in 1991 by Jindˇrich Neˇcas, with
the help of his then young students, amongst which Eduard Feireisel, Josef M´alek,
Antony Novotny, Mirko Rokyta, and Michael R˚uˇziˇcka, which are today active orga-
nizers, as well as well-known mathematicians. Eventually, Paseky’s school rapidly
established its reputation as one of the leading seminars for mathematical Fluid
Mechanics, and I was happy to give my contribution to the ninth edition.
A few words about the odd title of this paper, and on how such a paper finds its
place in the context of a school in Fluid Mechanics. The aim of my lectures was to
present a basic introduction to certain classical regularity issues, and to a few, new
regularization techniques that recently emerged in order to treat some variational
[5, 6, 66], and also non-variational [8], problems, whose non-standard structure
intervene also in the setting of Non-Newtonian Fluid Mechanics [274]. Originally
developed in a variational context, such methods were adapted, and extended to
approach more general problems [7, 10]. Therefore I decided to present a selection
of results and techniques, especially referring to the variational case, that in turn
should also apply, modulo suitable re-adaptations, to non-variational situations.
All this reflects the content of this paper; moreover, here I will also try to address
a few open problems. Such open problems will be often emphasized in the way the
words you are reading now are.
Date: January 15, 2006.
1

2 GIUSEPPE MINGIONE
Why talking about a “Dark Side”? It is my - maybe wrong - impression, that
today regularity problems are not as popular as they were once. Regularity meth-
ods are sometimes not very intuitive, and often overburdened by a lot of technical
complications, eventually covering the main, basic ideas. The well known motto:
“God (or Devil) is in the details” (of the estimates!), heavily applies here. More-
over, very often no room for partial results is given: either the whole problem is
solved, or really nothing comes up! So, fewer and fewer young analysts move to
face such issues, and regularity, especially in the Calculus of Variations, turns out
to be in the Dark Side. This paper aims to be a “friendly invitation” to come
to the Dark Side [229]. It collects some recent and non-recent regularity results
for minimizers of variational integrals, and solutions to elliptic systems/equations,
striving for casting a relatively general panorama in the unconstrained minimiza-
tion problem case. I shall start from the by now classical stuff, mostly developed
until the end of the eighties, and then I will come to some more recent material.
Of course the outcome will be unavoidably partial, strongly influenced by what has
been my personal research up to now, and I apologize for all that fine material
which will not find its room here, together with missed quotations of important
contributions. Nevertheless, I hope the reader will take up my invitation to the
Dark Side, eventually also hoping for some final redemption!
Acknowledgments. Of course I’d like to thank the organizers of the ninth
Paseky School for the invitation to give lectures, and to write this survey paper.
I would like to thank professors Stefan Hildebrandt (Bonn) and Jan Kristensen
(Oxford), for fruitful comments on an earlier version of the paper, and professor
Tadeusz Iwaniec (Syracuse, N. Y.) for the interest in the same earlier version. I also
thank the referees, for the remarks that eventually lead to an improved presentation,
and in particular, to that referee who detected in a precise way the “Star Wars saga”
references. I have been partially supported by MIUR via the project “Calcolo delle
Variazioni” (Cofin 2004), and by GNAMPA via the project “Studio delle singolarit`a
in problemi geometrici e variazionali”.
2. The scalar case, and the phantom irregularity
The results presented in this section can be considered as classical, and their
final settling, in most of the cases, dates back to the end of the eighties. Let me
start considering variational integrals of the type
(2.1) F(v, A) :=
Z
A
F (x, v, Dv) dx
defined for Sobolev maps v ∈ W
1,p
loc
(Ω, R
N
), and open sets A whose closure is
compact and contained in Ω. Here n ≥ 2, N ≥ 1, Ω is a bounded open set in R
n
,
p ≥ 1, and F : Ω × R
N
× R
nN
→ R is an integrand, for simplicity assumed to be
measurable with respect to the first variable, and continuous with respect to the
last two ones. In the following I will also denote
F ≡ F(v) ≡ F(v, Ω) .
A local minimizer of the functional F is a map u ∈ W
1,p
loc
(Ω, R
N
) such that
F(u, A) ≤ F(v, A) ,
whenever A ⊂⊂ Ω and u − v ∈ W
1,p
0
(A, R
N
). A classical problem in the Calculus
of Variations consists of studying the regularity properties of such maps.
Strongly connected to this problem is the one of regularity of weak solutions to
elliptic systems of the type
(2.2) div a(x, u, Du) = b(x, u, Du) ,

REGULARITY IN THE DARK SIDE 3
where a : Ω×R
N
×R
nN
→ R
nN
and b : Ω×R
N
×R
nN
→ R
N
are vector fields, again
assumed to be measurable with respect to the first variable, and continuous with
respect to the last twos. Indeed, when the integrand F (x, v, z) of F in (2.1) is regular
enough, minimizers of F solve the so called Euler-Lagrange system associated to F
(2.3) div F
z
(x, u, Du) = F
v
(x, u, Du) ,
which turns out to be elliptic provided F (x, v, z) satisfies suitable convexity as-
sumptions with respect to z, see [152], Chapters 1 and 2. The symbol F
z
denotes
of course the partial derivative of F with respect to the gradient variable z. A good
reference for regularity results for elliptic systems is also [31].
In the whole paper I will present a list of theorems and results, almost never
under the most general assumptions; I will rather prefer to confine myself to the
simplest, basic cases, in order to emphasize the main ideas. The interested reader
will find more material, and results under optimal assumptions, in the references
that I am going to provide through.
In this paper, I shall usually adopt the following viewpoint: given a local mini-
mizer u ∈ W
1,p
loc
(Ω, R
N
) of the functional F, or a weak solution to (2.2), what are
the additional regularity properties of u, in the interior of Ω? So, I will not discuss,
for instance, the regularity of u up to the boundary; that’s also why no assumptions
are made on the smoothness of ∂Ω in what follows. More importantly, I will not
address existence problems: for these the reader may look at, for example, the two
books by Dacorogna [72] and Giusti [165], as far as minimizers are concerned; for
equations and systems of the type (2.2), the well-known monotone operators theory
generally applies [228]. Giusti’s book is also a very goo d and smooth introduction
to some of the regularity topics I am going to deal with in the present review.
In this section I shall focus on the scalar case N = 1, later on I shall deal with the
vectorial one, that is N > 1 (sometimes I will refer to the vectorial case indicating
N ≥ 1, when I will present results valid for both the two cases). In the scalar case
both (2.2) and (2.3), whenever F is smooth enough to ensure that this last one
exists, become nonlinear elliptic equations in divergence form. A classical reference
for these is of course [213]. We shall see that in the scalar case, under suitable
assumptions on the integrand F (x, v, z) in (2.1) and the vector field a(x, v, z) in
(2.2), it is possible to build a satisfying regularity theory, and irregularity of minima
and solutions remains a phantom menace.
Let me fix an important notation here: in the rest of the paper ν, L and p will
denote three real numbers such that
0 < ν ≤ L < ∞ , p > 1 .
2.1 H¨older regularity. I shall start by the “following important fundamental
result” of De Giorgi [77], so defined in Morrey’s review [252]. Let me consider a lin-
ear elliptic equation in divergence form, with bounded and measurable coefficients:
div(a
i,j
(x)D
j
u) = 0, that is, in its weak formulation
(2.4)
Z
Ω
a
i,j
(x)D
j
uD
i
ϕ dx = 0 , ∀ ϕ ∈ C
∞
c
(Ω) .
The equation has bounded and elliptic coefficients {a
i,j
(x)}, which are nevertheless
supposed to be only measurable:
(2.5) |a
i,j
(x)| ≤ L , a
i,j
(x)λ
i
λ
j
≥ ν|λ|
2
,
for a.e. every x ∈ Ω, and every λ ∈ R
n
. In the rest of the paper I will go on using
the usual summation convention on repeated indexes. It is clear that the role of ν
is that of a lower bound for the eigenvalues of the matrix {a
i,j
(x)}, while L acts as
an upper one. We have

4 GIUSEPPE MINGIONE
Theorem 2.1. (De Giorgi). Let u ∈ W
1,2
(Ω) be a weak solution to the equa-
tion (2.4), under the assumptions (2.5). Then there exists a positive number
α ≡ α(n, L/ν) > 0 such that u ∈ C
0,α
loc
(Ω).
The previous result was independently obtained by Nash [264], directly for par-
abolic equations. Anyway, Nash’s techniques have not led to the massive develop-
ments of De Giorgi’s: see the comments in the introduction of [215], a book where
De Giorgi’s methods are extended to the parabolic case in great extent; but see
also the paper [117], where Nash’s techniques are revitalized. A little later Moser
[258, 259, 260] gave different proofs of De Giorgi’s and Nash’s results, proving actu-
ally the validity of Harnack’s inequality for solutions to (2.4), and to its parabolic
analog
Z
Ω×[0,T )
uϕ
t
− a
i,j
(x, t)D
j
uD
i
ϕ dx dt = 0 , ∀ ϕ ∈ C
∞
c
(Ω × [0, T )) .
Indeed, Harnack’s inequality implies in turn the local H¨older continuity of solutions,
see [165], notes to Chapter 7. Nowadays such basic regularity methods are indeed
known as De Giorgi-Nash-Moser’s theory. For an original and elegant approach to
such theory see also [31], Chapter 2.
In Theorem 2.1 the dependence of the H¨older continuity exponent α is critical
with respect to the “ellipticity ratio” L/ν of the matrix {a
i,j
(x)}:
(2.6) lim
L/ν→∞
α = 0 .
Indeed, the main strength of Theorem 2.1 is in the fact that the coefficients {a
i,j
(x)}
are allowed to be merely measurable; in the case of continuous coefficients, the re-
sult was in fact already known, and proved via perturbation metho ds, i.e. the so
called Korn’s trick; see also Schauder estimates techniques in Chapter 6 of [164].
In this case the solution turns out to be actually locally H¨older continuous with
any exponent α < 1, an effect of the continuity of coefficients. The difference of
Theorem 2.1 with respect to the continuous coefficients case is emphasized by (2.6),
which reflects the basic role of the sole ellipticity and growth assumptions (2.5).
The importance of De Giorgi’s theorem and technique is manifold: De Giorgi was
initially motivated to prove it, apparently after discussions with Stampacchia, by
the aim of solving the famous Hilbert’s 19th problem; I refer to the survey of Mar-
cellini [240] for an updated discussion of it, and to the older survey of Stampacchia
[286]. Even more imp ortantly, as we shall see in a few lines, De Giorgi’s insights
opened the way to the nonlinear theory, and they are are the cornerstone of what
is nowadays called “Non-linear Potential Theory”, see [181, 230, 227, 300].
De Giorgi’s proof rested on a then completely new method. Roughly speaking, it
is based on the idea of proving regularity properties of solutions via the analysis of
the decay and density properties of their level sets, a method that eventually became
pervasive in the whole regularity theory. Indeed De Giorgi’s proof starts with the
observation that a weak solution to (2.4) satisfies the following “Caccioppoli type
inequalities” [50] on level sets:
(2.7)



R
A(k,%)
|Du|
2
dx ≤
c
(R−%)
2
R
A(k,R)
|(u − k)
+
|
2
dx ,
R
B(k,%)
|Du|
2
dx ≤
c
(R−%)
2
R
B(k,R)
|(k − u)
−
|
2
dx .
Here 0 < % < R, and with s > 0
A(k, s) := {x ∈ B
s
: u(x) ≥ k} , B(k, s) := {x ∈ B
s
: u(x) ≤ k} ,
with B
s
⊂⊂ Ω denoting a ball of radius s, while (u − k)
+
:= max{u − k, 0} and
(k − u)
−
:= max{k − u, 0}. The constant c depends essentially on the ellipticity

REGULARITY IN THE DARK SIDE 5
ratio L/ν. From this only information, via an innovative iteration procedure, De
Giorgi was able to derive the H¨older continuity of solutions, with an exponent
depending on c, and therefore ultimately on L/ν. So, the whole H¨older continuity
information of solutions is encoded in the two inequalities (2.7); this motivates the
nowadays common definition stating that a function u is in the De Giorgi’s class
DG iff satisfies (2.7), for all possible choices of k, % and R. Extensions, and a gentle
introduction to De Giorgi’s method, can be found in [165], Chapter 7, and [230],
Chapter 2. See also the original papers [253, 285, 279, 296, 297, 299], and again the
monograph [213], where the original De Giorgi’s and Moser’s methods have been
deeply extended and clarified.
It was soon recognized that the linearity of the equation (2.4) played actually no
role in the proof of (2.7), the ideas involved being genuinely non-linear ones, and
the result was rapidly extended to a vast class general nonlinear elliptic equations
in divergence form [213]. The following result is an example. Let me consider an
elliptic equation of the type
(2.8) div a(x, u, Du) = 0 ,
under the following growth and monotonicity assumptions:
(2.9) |a(x, v, z)| ≤ L(1 + |z|
p−1
) , ν|z|
p
− L ≤ ha(x, v, z), zi ,
for every x ∈ Ω, v ∈ R, and z ∈ R
n
, with p > 1. Then we have
Theorem 2.2. Let u ∈ W
1,p
(Ω) be a weak solution to the equation (2.8), under
the assumptions (2.9). Then there exists a positive number α ≡ α(n, p, L/ν) > 0
such that u ∈ C
0,α
loc
(Ω).
Once again the proof, see for instance [230], rests on proving that a solution sat-
isfies inequalities similar to the ones in (2.7), with the growth exponent p replacing
2, and then applying De Giorgi’s method; the dependence of α is the same as the
one in (2.6). No pointwise regularity property of the vector field a is required with
respect to the variables (x, v). Extensions are possible to complete equations of the
type (2.2), including lower order terms in the formulation of the assumptions (2.9),
see [213] or [230]. For the extension of such result to parabolic equations of the
form
(2.10) u
t
− div a(x, u, Du) = 0 ,
I refer to [215] and, especially for the degenerate case including the evolutionary p-
Laplecean equation u
t
−div (|Du|
p−2
Du) = 0, to DiBenedetto’s book [81]. Such an
extension to degenerate problems is highly non-trivial, and involves DiBenedetto’s
innovative method of intrinsic geometry: using, in the formulation of the (parabolic)
Caccioppoli’s estimates, parabolic cylinder whose size is determined by the solution
itself.
Of course it was immediately observed that, for functionals of the type (2.1)
possessing the Euler-Lagrange equation (2.3) satisfying assumptions of the type
(2.9), the H¨older regularity of minimizers follows viewing them as solutions to
equations of the type (2.2). But it took not less than twenty years to start exploiting
the full impact of De Giorgi’s techniques on the regularity of minima. Indeed first
Freshe [132], under stronger assumptions, and then Giaquinta & Giusti [157], in full
generality, applied De Giorgi’s method to minimizers in a direct way, that is without
using the Euler-Lagrange equation, which may eventually not exist. More precisely,
considering only the following growth assumptions on the integrand F (x, v, z):
(2.11) ν|z|
p
≤ F (x, y, z) ≤ L(1 + |z|
p
) ,
we have

6 GIUSEPPE MINGIONE
Theorem 2.3. Let u ∈ W
1,p
(Ω) be a local minimizer of the functional F, under
the assumptions (2.9). Then there exists a positive number α ≡ α(n, p, L/ν) > 0
such that u ∈ C
0,α
loc
(Ω).
The proof in [157] is elegant and simple, and makes use of a clever application
of the hole-filling technique of Widman [304]. It essentially relies on the observa-
tion the sole minimality property, and growth conditions (2.11), force minimizers
to satisfy inequalities of the type in (2.7), with p replacing the exponent 2; then
H¨older continuity automatically follows via De Giorgi’s iteration method. There-
fore the result is valid for functions F (x, v, z) which are not differentiable with
respect to the v-variable, and even not convex with respect to the gradient vari-
able z; moreover, the result extends to the so-called ω-minima, see Paragraph 4.5
below. Shortly later, DiBenedetto & Trudinger [84] proved that minimizers also
satisfy the Harnack inequality under the assumptions of Theorem 2.3, thus extend-
ing to very irregular functionals Moser’s results for elliptic equations. Once again,
DiBenedetto & Trudinger directly proved the validity of Harnack’s inequality not
only for minima, but more generally for functions in De Giorgi’s classes DG. A full
extensions of such Harnack inequalities results to the general parabolic case (2.10)
is nevertheless still an open problem, see [80].
Coming back to the elliptic case, let me observe that De Giorgi’s techniques
open the way to study low order regularity also for equations and functionals with
coefficients in Lorentz spaces, see for instance [121].
2.2 Lipschitz type regularity. Up to now we have seen what are the assump-
tions implying the local H¨older continuity of minima and solutions to equations,
for some H¨older exponent α > 0. Now I will review some higher regularity results;
not surprisingly, in order to have higher regularity of solutions, and minimizers, one
must assume more regularity on the vector field a in (2.8), and on the integrand
F (x, v, z) in (2.1), as shown by the examples in [269, 221].
I shall start presenting an innovative result by Fonseca & Fusco [122, 123, 110],
concerning integral functionals of the type
(2.12) G
s
(v) :=
Z
Ω
ν|Dv|
p
+ g(Dv) dx .
Theorem 2.4. Let u ∈ W
1,p
(Ω) be a local minimizer of the functional G
s
, such
that g : R
n
→ R
+
is a convex function satisfying 0 ≤ g(z) ≤ L(1 + |z|
p
), for some
L ≥ 0. Then Du ∈ L
∞
loc
(Ω, R
n
).
The significance of this result lies in the fact that the regularity assertion is
this time made on the gradient Du, and this is usually achieved by using the
Euler-Lagrange equation of the functional, which has in turn to be differentiated
again. This allows to discover that the gradient itself is a solution to another
equation, and then to argue on this. In the case of the functional G
s
, the integrand
has the form F (z) = ν|z|
p
+ g(z), and the general assumptions on g only allow
to conclude that F is differentiable once, and only at almost every point, being
anyway a convex and therefore Lipschitz function. Fonseca & Fusco by-passed this
point combining essentially two ingredients: a delicate way of deriving a priori L
∞
-
estimates for the gradient Du when dealing with more regular integrands, and a
suitable approximation argument in order to approximate the functional G
s
with a
sequence of smoother ones, possessing the Euler-Lagrange equation. Observe that
in Theorem 2.4 the number ν > 0 can be picked small at will without any loss of
regularity on Du. An extension to Theorem 2.4 to more general functionals of the
type
(2.13) G
s
(v) :=
Z
Ω
ν|Dv|
p
+ g(x, v, Dv) dx ,

REGULARITY IN THE DARK SIDE 7
is possible, this time requiring in addition that g is continuous with respect to the
variable (x, v), uniformly with respect to z, that is
|g(x, u, z) − g(y, v, z)| ≤ Lω(|x − y| + |u − v|)(1 + |z|
p
) ,
for every x, y ∈ Ω, u, v ∈ R and z ∈ R
n
. Here ω : [0, ∞) → [0, 1] is a non-decreasing
continuous function vanishing at zero, what from now on I shall call a “modulus of
continuity”. In this case we have that u ∈ C
0,α
loc
(Ω), for every α < 1. This result has
been achieved in [68], and requires a further refinement of the techniques in [122],
based on the so called Ekeland’s variational principle [105]. The use of this last tool
to attack regularity problems was first introduced, in a different context, by Fusco
& Hutchinson in [142]; it eventually became a standard. It is to note that now α
is indep endent of the ratio L/ν, unlike in (2.6); this is the combined effect of the
global, strict convexity with respect to z exhibited by F (x, v, z) = ν|z|
p
+ g (x, v, z),
and the fact that the dependence on (x, v) is not just measurable, but now, rather,
continuous. Further extensions are in [70].
2.3 C
1,α
-regularity. I will finally pass to examine situations where higher reg-
ularity can be achieved. I will confine myself to consider the local H¨older continuity
of the gradient of solutions and minima. In fact, this is the focal point of the the-
ory. Once this type of regularity is achieved, then higher regularity of solutions,
up to analyticity, can be obtained by well-known boot-strap methods; I shall not
dedicate space to this ultra-classical point, which is essentially based on the so
called Schauder estimates for linear elliptic systems and equations with variable
coefficients; the reader is referred for instance to [165], Chapter 10, for a neat and
elementary presentation.
The first result I am going to report on concerns integral functionals of the type
F. The assumptions will be this time
(2.14)



















z 7→ F (x, v, z) is C
2
ν|z|
p
≤ F (x, v, z) ≤ L(1 + |z|
p
)
ν(µ
2
+ |z|
2
)
p−2
2
|λ|
2
≤ hF
zz
(x, v, z)λ, λi ≤ L(µ
2
+ |z|
2
)
p−2
2
|λ|
2
|F (x, u, z) − F (y, v, z)| ≤ Lω(|x − y| + |u − v|)(1 + |z|
p
) ,
for all x, y ∈ Ω, u, v ∈ R and z, λ ∈ R
n
, where µ ∈ [0, 1] is a fixed constant and
ω : R
+
→ (0, 1) is a continuous, non-decreasing modulus of continuity, such that for
some α ∈ (0, 1),
(2.15) ω(s) ≤ s
α
.
Assumption (2.14)
3
describes a controlled, uniform convexity of the integrand F ,
via growth conditions imposed on the second derivatives F
zz
, which are once again
prescribed accordingly to the ones in (2.14)
2
. On the other hand, assumption
(2.14)
4
, together with (2.15), means that the integrand F is H¨older continuous with
respect to (x, v) with exponent α ∈ (0, 1), uniformly with resp ect to z. Note that the
H¨older continuity condition has been re-normalized taking into account the growth
conditions in (2.14)
2
. Roughly speaking, when prescribing (2.14), one thinks of
model examples such as F (x, v, z) ≡ c(x, v)(1 + |z|
2
)
p/2
, or F (x, v, z) ≡ c(x, v)f(z),
and ν ≤ c(x, v) ≤ L.
In the previous assumptions the parameter µ plays a very important role. When
µ > 0, the functional is non-degenerate elliptic. The case µ = 0 corresponds to
degenerate cases. For instance, a model case when µ > 0 is given by
Z
Ω
c(x, v)(1 + |Dv|
2
)
p
2
dx .

8 GIUSEPPE MINGIONE
where ν ≤ c(x, v) ≤ L is a H¨older continuous function; here of course µ = 1. A
typical degenerate model is
Z
Ω
c(x, v)|Dv|
p
dx .
In the case c(x, v) ≡ 1 we have the p-Dirichlet functional
(2.16) D
p
(v) :=
Z
Ω
|Dv|
p
dx ,
where of course we are taking µ = 0, and whose Euler-Lagrange equation is the
following well-known, degenerate p-Laplacean equation:
(2.17) div(|Du|
p−2
Du) = 0 .
Actually in (2.14) the only important cases are the degenerate one µ = 0, and
µ = 1, a case we can always reduce to when µ > 0, provided we increase the ratio
L/ν enough, depending on the how µ is close to 0.
I shall present two theorems; the first concerns the non-degenerate case µ > 0:
Theorem 2.5. Let u ∈ W
1,p
(Ω) be a local minimizer of the functional F, under
the assumptions (2.14), with µ > 0. Then Du ∈ C
0,α/2
loc
(Ω, R
n
).
It is to be noted that the degree of regularity of F (x, v, z) with respect to (x, v),
directly influences the degree of regularity of the gradient. This is a well known
phenomenon, think of Schauder estimates for linear elliptic equations [164]. The
second result regards the degenerate case µ = 0.
Theorem 2.6. Let u ∈ W
1,p
(Ω) be a local minimizer of the functional F, un-
der the assumptions (2.14) with µ = 0. Then Du ∈ C
0,β
loc
(Ω, R
n
), for some β ≡
β(n, p, L, α) > 0.
In this last case there is a loss in the H¨older continuity degree of the gradient,
due to the fact that the problem is actually degenerate. This means the following,
looking at equation (2.17): letting a(z) := |z|
p−2
z, the lowest eigenvalue of the
matrix a
z
is comparable to |z|
p−2
. So, when |Du| approaches zero, the equation
(2.17) looses its ellipticity properties and estimates worsen. Indeed, even in the
case of solutions to (2.17), and minima of (2.16), the gradient Du is not β-H¨older
continuous with any exponent β < 1, as shown by Ural’tseva in 1968 [302]. On the
contrary, minima of
Z
Ω
(µ
2
+ |Dv|
2
)
p
2
dx , µ > 0 ,
are C
∞
(Ω): in this case the associated Euler-Lagrange equation is non-degenerate
elliptic. Theorem 2.6, and the present form of Theorem 2.5, are due to Manfredi
[231], a paper I refer to for further references on degenerate problems; the first
form of Theorem 2.5, under more restrictive assumptions, and in particular for
the case p ≥ 2, and in the non-degenerate case µ > 0, is independently due to
Giaquinta & Giusti [158, 159], and Ivert [188]. Once again, Theorems 2.5, and
2.6 are significant because the regularity of the gradient is obtained for functionals
that do not necessarily possess the Euler-Lagrange equation (2.3); the functionals
considered are indeed non-differentiable since the dependence on the variable v
of the integrand F is just H¨older continuous, and therefore F
v
does not exists in
general. This was the main contribution in [188, 158].
Theorems 2.5 and 2.6 have their counterparts for elliptic equations. For sim-
plicity I shall report on the homogeneous case (2.8). Let me consider the following

REGULARITY IN THE DARK SIDE 9
assumptions, which are the natural reformulation of the ones in (2.14) when arguing
on the vector field a, rather than on the integrand F:
(2.18)



















z 7→ a(x, v, z) is C
1
|a(x, v, z)| ≤ L(1 + |z|
p−1
)
ν(µ
2
+ |z|
2
)
p−2
2
|λ|
2
≤ ha
z
(x, v, z)λ, λi ≤ L(µ
2
+ |z|
2
)
p−2
2
|λ|
2
|a(x, u, z) − a(y , v, z)| ≤ Lω(|x − y| + |u − v|)(1 + |z|
p−1
) ,
for all x, y ∈ Ω, u, v ∈ R and z, λ ∈ R
n
, where µ ∈ [0, 1] is a fixed constant and
ω : R
+
→ (0, 1) is as in (2.15). Then we have the natural analogs of Theorems 2.5
and (2.6):
Theorem 2.7. Let u ∈ W
1,p
(Ω) be a weak solution to the equation (2.8), under
the assumptions (2.18), with µ > 0. Then Du ∈ C
0,α
loc
(Ω, R
n
).
As for Theorem 2.7 compared to Theorem 2.5, let me notice that the degree of
H¨older continuity of the gradient increases, from α/2 to α. This is a general prin-
ciple: minimality itself is a property strong enough in order to guarantee regularity
also when dealing with irregular, non-differentiable functionals, but the property
of satisfying an equation is stronger and forces higher regularity. The degenerate
analog is finally the following:
Theorem 2.8. Let u ∈ W
1,p
(Ω) be a weak solution to the equation (2.8), un-
der the assumptions (2.18) with µ = 0. Then Du ∈ C
0,β
loc
(Ω, R
n
), for some β ≡
β(n, p, L, α) > 0.
The proof of Theorems 2.5-2.8 is based on a comparison argument using the so
called freezing method. In their full, “degenerate” generality, Theorems 2.7 and 2.8
are due to Manfredi [231]. Unfortunately, many of the interesting estimates Man-
fredi developed are not included in the paper [231], but are nevertheless retrievable
in his Ph.D. thesis [232], that for the clearness of exposition I highly recommend
as a first approach to the subject. For degenerate elliptic problems in the vectorial
case, and further techniques, see [143, 171].
3. The revenge of irregularity: the vectorial case
There were still hopes for getting a vectorial version of De Giorgi’s Theorem 2.1
around 1967, when De Giorgi himself showed that no such extension would take
place.
3.1 De Giorgi’s example [79]. This actually deals with functionals, and there-
fore also simultaneously shows that no extension to Theorems 2.1 and 2.3 can be
achieved when N > 1. The counterexample for systems follows of course by consid-
ering the corresponding Euler-Lagrange system. De Giorgi considered the following
quadratic-type functional with discontinuous coefficients:
(3.1) DG :=
Z
B
1
|Du|
2
+


(n − 2)
n
X
i=1
D
i
u
i
+ n
n
X
i,j=1
x
i
x
j
|x|
2
D
i
u
j


2
dx , n = N .
For n ≥ 3 the map
(3.2) u(x) :=
x
|x|
α
, α :=
n
2
"
1 −
1
p
(2n − 2)
2
+ 1
#
,
belongs to W
1,2
(B
1
, R
n
), and locally minimizes DG. Let me just observe that in this
wonderfully simple example it is necessary to take discontinuous dependence on x

10 GIUSEPPE MINGIONE
in the integrand, otherwise solution are H¨older continuous, and with any exponent
α < 1, by perturbation methods. No such example is possible when n = 2, since
in this case H¨older continuity of solutions, for some exponent α < 1, follows by the
higher integrability of Theorem 4.1 below, and Sobolev’s embedding Theorem. Let
me point out that almost independently and the same time, Maz’ya [247] found
an example of higher order elliptic equations, with analytic coefficients, but with
discontinuous solutions.
3.2 Giusti & Miranda’s example [166]. The main point in De Giorgi’s exam-
ple is the singularity of the matrix {a
α,β
i,j
(x)} at the origin. When the coefficients
matrix depends on the solution Giusti & Miranda showed that the matrix {a
α,β
i,j
(v)}
can be even analytic. They considered the quadratic-type functional
(3.3) GM(v) :=
Z
B
1
a
α,β
i,j
(v)D
j
v
α
D
i
v
β
dx , n = N ,
with
a
α,β
i,j
(v) := δ
i,j
δ
α,β
+
·
δ
α,i
+
4
n − 2
·
v
α
v
i
1 + |v|
2
¸
·
·
δ
β,j
+
4
n − 2
·
v
β
v
j
1 + |v|
2
¸
.
Here δ
i,j
denotes the usual Kronecker’s symbol. For n > 2 sufficiently large, the
discontinuous map
(3.4) u(x) :=
x
|x|
,
locally minimizes GM. Similar examples also work for quasilinear systems of the
type
(3.5)
Z
Ω
a
α,β
i,j
(u)D
j
u
α
D
i
ϕ
β
dx = 0 , ∀ ϕ ∈ C
∞
c
(Ω, R
n
) .
The key to understand why in this example analytic coefficients are allowed, with
respect to the one in Paragraph 3.1, is that if we think of the identification b
α,β
i,j
(x) ≡
a
α,β
i,j
(u(x)), then the map x → b
α,β
i,j
(x) is just measurable. Remarkable extensions
of De Giorgi’s and Giusti & Miranda’s constructions are given in [284, 194], where
nowhere continuous solutions to elliptic systems are produced via the construction
of a suitable, very pathological coefficient matrix {a
α,β
i,j
}.
3.3 Neˇcas’ example [265, 176]. In the previous examples the singularity of
minimizers occur by the peculiar way the discontinuous coefficients (x, v) mix-up
with the components of the gradient variable z. What happens when there are no
coefficients? This question was first answered by Neˇcas, who considered a simple
functional of the type
(3.6) F
s
(v) :=
Z
Ω
F (Dv) dx ,
and whose example immediately applies to systems when considering the Euler-
Lagrange system associated to F
s
(3.7) div F
z
(Du) = 0 .
In Neˇcas’ example the integrand F : R
nN
→ R
+
is analytic, with quadratic growth,
and satisfies the uniform ellipticity and growth conditions
(3.8) ν|λ|
2
≤ hF
zz
(z)λ, λi ≤ L|λ|
2
,
for all z, λ ∈ R
nN
. The integrand F (z) is rather complicated, and it can be found
in [176], formula (3.1). With Ω ≡ B
1
⊂ R
N
, the minimizer considered this time is
the map u : B
1
→ R
n
2
defined by
(3.9) u
i,j
(x) :=
x
i
x
j
|x|
−
1
n
δ
i,j
|x| .

REGULARITY IN THE DARK SIDE 11
The importance of this example also lies in the fact that it shows that the irreg-
ularity of minima is peculiar of the vectorial case, and is not due to the presence
of coefficients. Neˇcas’ example only works for n ≥ 5, while in the case of (non-
minimizing) solutions to systems an example exists starting with n ≥ 3 [266]. Note
that the map in (3.9) is not C
1
but still Lipschitz continuous, therefore the ex-
ample relates to Theorem 2.5, but not to Theorem 2.3. The problem of finding
non-Lipschitz minimizers remained an important open issue in the theory for more
than 25 years. . .
3.4
ˇ
Sver´ak & Yan’s example [290, 291]. . . . until it was settled by
ˇ
Sver´ak &
Yan, who showed that minimizers of analytic functionals as considered by Neˇcas,
and therefore also satisfying (3.8), maybe even unbounded! The construction offered
in [291] does not produce an explicit formula for the integrand F (z); the minimizer
u : B
1
→ R
n
2
considered this time is a variant of Neˇcas’ in the sense that
u
i,j
(x) :=
1
|x|
ε
µ
x
i
x
j
|x|
−
1
n
δ
i,j
|x|
¶
,
for suitable, positive values of ε. Moreover,
ˇ
Sver´ak & Yan also offer a complete
new example of a non-Lipschitz minimizer in low dimensions, taking n = 4 and
N = 3, and they also give an example of a non-Lipschitz minimizer when n = 3
and N = 5. It still remains the open problem to find a non-C
1
minimizer for the
case n = N = 3, which plays a role in the applications. Note anyway that, by higher
differentiability and integrability of solutions, and Sobolev’s embedding Theorem,
minima are H¨older continuous up to dimension n = 4, and in this sense
ˇ
Sver´ak
& Yan’s example of unbounded minimizers is dimensionally optimal; see the first
section of [291] for further comments.
4. A new hope: partial regularity
We have just seen that in the general vectorial case N ≥ 1 both minima of
variational integrals and solutions to elliptic systems may develop singularities.
Moreover, at least by looking at the phenomena observed up to now, we may say
that everywhere regularity occurs very rarely in the vectorial case. On the contrary,
especially in geometrically constrained problem, it is possible to see that minimizing
the energy naturally creates singularities. For instance, taking as Ω ≡ B
1
⊂ R
n
, the
map x 7→ x/|x|, already met in (3.4), minimizes the functional D
p
in (2.16) in its
Dirichlet class when p < n, amongst all the maps taking values in the unit sphere of
R
n
. This happens for very topological reasons. See the most recent contributions
[178, 186], and related references. For such issues I recommend to give a look at
[177].
4.1 Higher integrability. Anyway, a few, weaker forms of regularity still
persist. The only global regularity property, surviving in general to the passage
from the scalar to the vectorial case, is the called “higher integrability”, and it is
the content of the following:
Theorem 4.1. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F,
under the assumption (2.11). Then there exists a positive number s(n, p, L/ν) > p
such that Du ∈ L
s
loc
(Ω, R
nN
).
The previous theorem has b een obtained first by Attouch & Sbordone in a par-
ticular situation [22], and then by Giaquinta & Giusti [157] in the full generality
considered here. In the previous statement, as in the rest of the section, z ∈ R
nN
,
and whenever considered, all the assumptions stated in Section 2 must be recast
keeping into account this fact, and that now u : Ω → R
N
.

12 GIUSEPPE MINGIONE
The proof of the previous theorem is based again on Caccioppoli’s inequality, in
the form:
Z
B
R/2
|Du|
p
dx ≤
c
R
p
−
Z
B
R
|u − u
R
|
p
dx + cR
n
for ball B
R
⊂⊂ Ω, and u
R
is the average of u over B
R
; here c ≡ c(L/ν). Then one
applies Poincar`e inequality to get
−
Z
B
R/2
|Du|
p
dx ≤
µ
−
Z
B
R
|Du|
np
n+p
+ 1 dx
¶
n+p
n
,
obtaining what it is usually called a Reverse-H¨older inequality with increasing
support. At this point one uses Gehring’s lemma in one of its local versions
[288, 161, 44], and concludes with the existence of an higher integrability exponent
s ≡ s(L/ν) > p, such that
Ã
−
Z
B
R/2
|Du|
s
dx
!
1
s
≤
µ
−
Z
B
R
|Du|
p
+ 1 dx
¶
1
p
.
In other words, when passing to the vectorial case, Caccioppoli inequalities are still
able to provide some regularity, in the form of higher integrability. Needless to say
the same applies to systems:
Theorem 4.2. Let u ∈ W
1,p
(Ω, R
N
) be a weak solution to the system (2.8), under
the assumptions (2.9). Then there exists a positive number s(n, p, L/ν) > p such
that Du ∈ L
s
loc
(Ω, R
nN
).
Concerning the exponent s from Theorems 4.1 and 4.2, this is explicitly com-
putable, in the sense that lower estimates for it are available using the ones for
Gehring’s lemma [44, 197]. Anyway sharp bounds in terms of the ellipticity ratio
L/ν are not known, but for two dimensional, linear elliptic equations [221], when
they rest on a deep theorem of Astala [21], see also [193]. A very interesting ex-
tension of Theorem 4.2 to the case of parabolic systems with p-growth has been
achieved by Kinnunen & Lewis in [198]; for the case p = 2 see [163].
Reverse H¨older’s inequalities and higher integrability results were first obtained
by Gehring for quasi-conformal mappings in his epoch-making paper [150], while
the application to higher order equations and systems was obtained by Elcrat &
Meyers [106]. Local extensions, suitable for further applications to regularity prob-
lems, were obtained by Giaquinta & Modica [161], and Stredulinsky [288]. I also
recommend to give a look at the very nice proof given by Bojarski & T. Iwaniec [44];
extensions to the setting of Orlicz spaces, and in certain limit function spaces, are
also available [146, 148, 191, 192, 120, 41]. For further properties and information
concerning Reverse H¨older inequalities and higher integrability, I again recommend
the nice surveys by T. Iwaniec [191], and Sbordone [277], and the thesis of Kinnunen
[197], where a detailed study of the various constants occurring in Reverse H¨older
inequalities is cleverly carried out. Related, previous results, are in the works of
Bojarski & Sbordone & Wik [45], and D’Apuzzo & Sbordone [75]. For connections
to Harmonic Analysis I recommend the paper [67] and its references.
4.2 Partial C
1,α
-regularity. Concerning the pointwise regularity (in the in-
terior of Ω) of minima and solutions, the so called partial regularity comes into
the play. The general principle of partial regularity asserts the pointwise regularity
of solutions/minimizers, in open subset whose complement is negligible. In other
words, one tries to prove that the solution, or the minimizer u, is regular, in some
specified sense, in an open subset Ω
u
⊂ Ω such that |Ω \ Ω
u
| = 0; the set
(4.1) Σ
u
:= Ω \ Ω
u
,

REGULARITY IN THE DARK SIDE 13
is called the singular set of u. For this reason partial regularity is sometimes called
almost everywhere regularity.
The first instance of such approach I am presenting is given by the following
partial regularity analog of Theorem 2.5:
Theorem 4.3. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F,
under the assumptions (2.14) with µ > 0. Then there exists an open subset Ω
u
⊂ Ω
such that |Ω \ Ω
u
| = 0, and Du ∈ C
0,α/2
loc
(Ω
u
, R
nN
).
The analog of Theorem 2.7 is instead the following:
Theorem 4.4. Let u ∈ W
1,p
(Ω, R
N
) be a weak solution to the system (2.8), under
the assumptions (2.18) with µ > 0. Then there exists an open subset Ω
u
⊂ Ω such
that |Ω \ Ω
u
| = 0, and Du ∈ C
0,α
loc
(Ω
u
, R
nN
).
The previous two theorems have no analog in the degenerate case µ = 0, unless
some further structure assumptions are added, as we shall see in Paragraph 4.9
below. The singular set Σ
u
= Ω \ Ω
u
in such theorems is identified by the equality
(4.2) Σ
u
= Σ
0
u
∪ Σ
1
u
,
where
Σ
0
u
:=
(
x ∈ Ω : lim inf
ρ&0
−
Z
B(x,ρ)
|Du(y) − (Du)
x,ρ
|
p
dy > 0
or lim sup
ρ&0
|(Du)
x,ρ
| = ∞
)
,
Σ
1
u
:=
(
x ∈ Ω : lim inf
ρ&0
−
Z
B(x,ρ)
|u(y) − (u)
x,ρ
|
p
dy > 0
or lim sup
ρ&0
|(u)
x,ρ
| = ∞
)
.
The reason for (4.2) is very basic: the partial regularity technique leading to The-
orems 4.3 and 4.4 is actually a linearization technique. Let me sketch it in a simple
case, the one of systems of the type div a(D u) = 0, when we actually have Σ
u
= Σ
0
u
.
Looking at the proofs, one realizes that the regular points x, that is the points the
gradient is H¨older continuous in a neighborhood of, are basically those ones such
that, for a some radius r > 0
(4.3) −
Z
B(x,r)
|Du(y) − (Du)
x,r
|
p
dy ≤ ε ,
and ε > 0 is suitably small. A regularity condition as (4.3) is usually referred to
as an “ε-regularity criterium”. The original solution u is compared, in the ball
B
r
≡ B(x, r), with the solution v ∈ W
1,p
(B
r
, R
N
) of the auxiliary, linearized sys-
tem div(D
z
a((Du)
x,r
)Dv) = 0, which is a linear elliptic systems with constant
coefficients. Therefore the comparison map v is smooth, and enjoys good a-priori
estimates. Here ( Du)
x,r
is the average of Du over B
r
. Then the next step is to
make sure that u and v are, in some integral sense, near enough in order to make
u inherit the regularity estimates of v. This is achieved if the original system is
“near enough” to the linearized one div(D
z
a((Du)
x,r
)Dv) = 0. Condition (4.3)
serves to ensure this. From this argument the characterization in (4.3) naturally
pops up, as well as the identity in (4.2). Such a linearization idea finds its origins
in Geometric Measure Theory, and more precisely in the pioneering work of De
Giorgi [78] on minimal surfaces, and of Almgren [12] for minimizing varifolds, and
was first implemented by Morrey [256], and Giusti & Miranda [167], for the case

14 GIUSEPPE MINGIONE
of quasilinear systems div(a(u)Du) = 0. Great impulse to the study of partial
regularity of solutions to systems, and minima of functionals, was initially given
by the study of harmonic mappings and related elliptic systems, carried out in the
papers by Hildebrandt & Kaul & Widman [182, 183]. For the completely non-
linear case we have today different methods to implement the local linearization
scheme described above: the hard, “direct method” applied by Giaquinta & Mod-
ica [160] and Ivert [187, 188]; the indirect one via blow-up techniques, implemented
originally in the cited papers of Morrey and Giusti & Miranda, and then recov-
ered directly for the quasiconvex case by Evans, Acerbi, Fusco, Hutchinson, and
Hamburger [115, 142, 2, 173, 175]; see also Theorem 4.11, and comments, below.
Finally, the technique I like most, the “A-approximation method”, once again first
introduced in the setting of Geometric Measure Theory by Duzaar & Steffen [104],
and applied to partial regularity for elliptic systems and functionals by Duzaar &
Gastel & Grotowski [94, 92], see also the nice survey [97]. This method re-exploits
the original ideas that De Giorgi introduced in his treatment of minimal surfaces
[78], providing a neat and elementary proof of partial regularity. It also shows that
the heavy tool of Reverse H¨older inequalities originally used in the papers of Gi-
aquinta & Modica and Ivert, can be actually completely avoided. The linearization
is indeed implemented via a suitable variant, for systems with constant coefficients,
of the classical “Harmonic approximation Lemma” of De Giorgi, see also [283] and
Paragraph 4.9 below. The foregoing rough explanation also suggests why we have
no analog of Theorem 4.4 in the case µ = 0: when linearizing the system about
the gradient average (Du)
x,r
, it may happen that (Du)
x,r
is near the origin, or
even zero, so that the linearized systems itself looses its ellipticity and regularizing
properties, and at the end no comparison argument takes place. In such degenerate
cases more accurate comparison procedures must be followed, and under additional
structure assumptions on the integrand F , see Paragraph 4.9 below.
4.3 Lack of low order partial regularity. Although someone may think
differently, partial regularity theory for both systems and functionals is still widely
incomplete; a whole partial regularity analog of the low regularity theory, for in-
stance in the spirit of Paragraph 2.2, is yet missing. Let me mention the following
problem, which is amazingly still open: consider an elliptic system of the simple
type
div a(x, Du) = 0 ,
satisfying (2.18), where this time we are not requiring (2.15), but we are just asking
ω(·) to vanish at zero. In other words, the dependence of a(x, z) upon x is just
continuous, rather than H¨older continuous. It is natural to ask whether there exists
an open subset Ω
0
⊂ Ω such that |Ω \ Ω
u
| = 0, and such that u ∈ C
0,α
loc
(Ω
0
, R
N
),
for some α > 0. Eventually for any α < 1. Note that in the case of equations, and
scalar functionals N = 1, this is known, with no singular set: Ω = Ω
u
; compare
Paragraph 2.2. The answer to such a basic issue is at the moment not known but
in certain low dimensional cases, as correctly shown by Campanato, see [52] and
the survey [55]. It is interesting to know that Campanato himself gave a “proof of
the result” in the general case, which revealed to be completely wrong [54]. More
generally, no partial regularity result of any type is known for solutions to such a
system. A similar problem poses of course for functionals, just considering (2.14),
without requiring (2.15).
4.4 The size of the singular set. After getting Theorems 4.3 and 4.4, the
next issue is of course trying to prove that Σ
u
is not only negligible, but in some
sense, “smaller”, if not empty at all. A way to do this is to give an upper estimate
for the Hausdorff dimension dim
H
(Σ
u
) of Σ
u
. For basic systems of the type
(4.4) div a(Du) = 0 ,

REGULARITY IN THE DARK SIDE 15
and therefore for simple functionals of the type (3.6), it is possible to prove that
(4.5) dim
H
(Σ
u
) < n − 2 .
See for instance [52]. In particular, when n = 2, the singular set is empty. In the
case of systems (2.8), assuming that both a
v
and a
x
exist, and that the solution u
is a-priori continuous, Giaquinta & Modica [160] proved (4.5) again. The problem
in the general case (2.10) remained open (stated in [152], page 191, and [160],
page 115) since [187, 160]. The first results in this direction can be found in my
papers [248, 249], that I will summarize now, also keeping into account of some
later improvements in [206, 98]. I shall start with the following result, essentially
contained in [249]:
Theorem 4.5. Let u ∈ W
1,p
(Ω, R
N
) be a weak solution to the system (2.8), under
the assumptions (2.18), with µ > 0. Then
(4.6) dim
H
(Σ
u
) ≤ n − min{2α, s − p} ,
where s > p is the higher integrability exponent appearing in Theorem 4.2. Moreover
if n ≤ p + 2, or if the solution u is already locally H¨older continuous in Ω, then
(4.7) dim
H
(Σ
u
) ≤ n − 2α .
Finally, when p = 2 the previous inequalities become strict and, in the second case,
H
n−2α
(Σ
u
) = 0.
As stated in Theorem 4.2, the number s can be explicitly quantified, essentially
in terms of the ellipticity ratio L/ν; see [44, 197, 288], and the discussion immedi-
ately after Theorem 4.2. Getting information on the Hausdorff dimension of certain
branch, or removable, sets, by estimating higher integrability exponents, is a strat-
egy typically followed in the theory of quasiconformal mappings, where the role of
the ration L/ν is played by the quasiconformality constant K of a quasiconformal
mapping f ∈ W
1,n
(Ω, R
n
):
|Df(x)|
n
≤ Kdet(Df(x)) ,
see [44, 190, 47]. In the last part of Theorem 4.5 the restriction to the case p = 2
appears to be technical, and I hope in the future someone will achieve the strict
inequality for any p > 1. Results are also available for complete systems of the type
in (2.2), see again [249]. There’s also a further result from [248]:
Theorem 4.6. Let u ∈ W
1,p
(Ω, R
N
) be a weak solution to the simpler system
(4.8) div a(x, Du) = 0 ,
under the assumptions (2.18) with µ > 0, suitably recast for such case. Then
(4.9) dim
H
(Σ
u
) ≤ n − 2α .
Finally, when p = 2 the previous inequality becomes strict, and H
n−2α
(Σ
u
) = 0.
Comments are in order. Let me start from the last theorem. Estimate (4.9)
tells us that the possibility of “reducing” the dimension of the singular sets is
determined in a quantitative way by the regularity with respect to the coefficients:
the more regular a(x, z) is with respect to x, the better the estimate becomes.
It is a sort of Schauder estimate for the singular set, and it agrees with (4.5),
obtained assuming, amongst the other things, differentiability with respect to x,
that is, roughly, α = 1. Such a viewpoint helps to give an intuitive explanation to
estimate (4.6), valid in the general case. When dealing with a complete vector field
of the type a(x, u, z), the system (2.8) can be viewed as div b(x, Du) = 0, where
b(x, z) ≡ a(x, u(x), z). At this point the H¨older continuity of x → b(x, z) is lost,
since u(x) may exhibit high irregularity, compare Paragraph 3.2. Nevertheless, the

16 GIUSEPPE MINGIONE
fact that u(x) is actually a solution to the system comes into the play again via
Theorem 4.2, and the L
s
−integrability of Du serves to bound in a suitable way the
oscillations of u(x). Accordingly, in the low dimensional case n ≤ p+2, it is possible
to prove that u is H¨older continuous outside a closed subset of Hausdorff dimension
less than n−2, and eventually we can recover the full estimate (4.7) again. The same
obviously applies when u(x) is a-priori assumed to be everywhere H¨older continuous.
The technique for proving estimates (4.6)-(4.9) rests on the simple observation that
the H¨older continuity dependence of the vector field a(x, u, z), can b e read as a
fractional differentiability. Therefore, applying a variant of the standard difference
quotients method technique [267], via suitable test functions, and in combination to
Gehring’s lemma, it is possible to prove that the gradient is in a suitable fractional
Sobolev space. In turn, this implies the estimate on the singular sets via abstract
measure theoretical arguments. For details see [248, 249]. Extensions to the case of
systems with Dini continuous coefficients, both with respect to partial regularity,
and to the singular sets dimension estimates, are also possible [91, 305, 93]. In
this case more general Hausdorff measures come into the play, and more precisely
those ones generated using Carath´eodory’s construction via a gage (generating)
function which is not of power type; see [93] and references for more. The results
of Theorems 4.5 and 4.6 have been extended by Kronz [212] to the case of higher
order elliptic systems.
The problem of estimating the Hausdorff dimension in the case of minima also
remained open since the papers [157, 158, 188], and even in the favorable case of
C
∞
dependence of the integrand F (x, v, z) with respect to (x, v); see the comments
below, after Theorem 4.8. It was not clear whether |Σ
u
| = 0 was already optimal
or not, and the issue was raised several times: see for instance [153], open problem
in 3, [154] comments in Section 4, and [156], open problem (a), page 117. This
problem has been settled by Kristensen & myself in [206], where we have proved
that partial regularity in the sense of |Σ
u
| = 0 is never optimal. The first result is
the analogue of Theorem 4.6:
Theorem 4.7. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F,
under the assumptions (2.14) with µ > 0. Then
(4.10) dim
H
(Σ
u
) ≤ n − min{α, s − p} ,
where s > p is the higher integrability exponent appearing in Theorem 4.1. More-
over, if n ≤ p + 2, we have
dim
H
(Σ
u
) ≤ n − α .
Finally, when p = 2 the previous inequalities become strict and, in the second case,
H
n−α
(Σ
u
) = 0.
Again, the number s essentially depends on L/ν, and therefore the bound on the
Hausdorff dimension given in (4.10) can be made explicit, and quantified in terms
of L/ν. Considerations analogous to those already made for the case of systems
apply here. Just observe that, exactly as in Theorem 4.3 with respect to Theorem
4.4, there is a loss in the estimate, n − α instead of n − 2α, when passing from
solutions to systems to minimizers of functionals. Nevertheless, the bound in (4.6)
can be recovered for minimizers in certain special cases [208]. It is interesting to
point out that for functionals with a special structure such as
(4.11)
Z
Ω
a(x, u)|Du|
p
dx ,

REGULARITY IN THE DARK SIDE 17
the bound can be improved up to n − p, see [157, 143]. It is also important to
note that, both for systems and functionals, the singular set is empty in the two-
dimensional case, at least when p ≥ 2; see Section 9 in [206], and comments in
Remark 2.2 from [98].
As far as the dependence on u is concerned, a sort of analog of the result of
Theorem 4.6 is available in the following:
Theorem 4.8. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional
(4.12)
Z
Ω
f(x, Dv) + g(x, v) dx ,
under the assumptions (2.14) on f, with µ > 0, and where g : Ω × R
N
→ R is
a bounded, measurable function which is α-H¨older continuous with respect to the
second variable, uniformly with respect to the first one. Then
(4.13) dim
H
(Σ
u
) ≤ n − α .
Finally, when p = 2 the previous inequality becomes strict, and H
n−α
(Σ
u
) = 0.
An explanation for the last improvement with respect to the estimate in (4.10),
is that now the “disturbing” presence of
u
(
x
) is decoupled by the regularizing term,
that is the one containing Du. It is interesting to see that g can be taken to be
only measurable in x; in this case partial regularity of minima has been proven
in [173]. The previous theorem is a particular case of the ones contained in [206],
where more general splitting structures of the type in (4.12) can be considered. The
idea for proving Theorems 4.7, and 4.8, is again showing that Du is in a fractional
Sobolev space, but this time the implementation must be completely different. In
fact, the functionals under consideration do not posses the related Euler-Lagrange
system, and it is not possible to use any test function technique. On the contrary,
in [206] we introduce a new, “variational difference quotients method”, based on
the minimality of u, and a delicate iteration/interpolation procedure in the setting
of Fractional Sobolev spaces. The basic idea is the following: since one cannot use
the Euler-Lagrange system of the functional, then one considers the Euler-Lagrange
systems of certain differentiable functionals, obtained from the original one by a
freezing procedure; in turn, these can be differentiated, and the related estimates
are transferred to the original minimizer by a comparison argument. The final effect
is an “indirect differentiation” of the original functional. Note that, as mentioned
above, even assuming C
∞
-regularity of F (x, v, z) the situation does not improve:
provided it does exist, the Euler-Lagrange system of the functional F is in general
a non-homogeneous, of the type (2.3), with a right hand side with critical growth
i.e.: |F
u
(x, u, Du)| ≤ L(1 + |Du|
p
). It is not possible to use the usual difference
quotients technique via test functions unless the solution is bounded, with suitably
small L
∞
-norm. On the other hand, this is not the case in general, as we have seen
in Paragraph 3.4.
It remains in my opinion an interesting open problem to discuss the optimality
of the Hausdorff dimension estimates contained in Theorems 4.5-4.8, eventually
finding minima and solutions with large singular sets.
There are of course plenty of further problems arising in the analysis of singular
sets: are they rectifiable? Do they have additional geometric structures? As for
the rectifiability of singular sets, in the case of harmonic maps, and of minimal
surfaces, the reader should give a look at the beautiful works of Simon [283, 282].
4.5 ω-minima, and their singular sets. The techniques introduced to treat
the singular set of non-differentiable functionals also apply to the so called ω-
minima. A map u ∈ W
1,p
(Ω, R
N
) is an ω-minimizer of the functional F in (2.1),

18 GIUSEPPE MINGIONE
under the assumption (2.11), if and only if
(4.14)
Z
B
R
F (x, u(x), Du(x)) dx ≤ [1 + ω(R)]
Z
B
R
F (x, v(x), Dv(x)) dx
for any v ∈ W
1,p
(B
R
, R
N
) such that u− v ∈ W
1,p
0
(B
R
, R
N
), where ω : R
+
→ R
+
is a
non-decreasing, concave function satisfying ω(0) = 0, and B
R
⊂⊂ Ω is an arbitrary
ball of radius R. ω-minima are sometimes called almost minimizers. Clearly, a
minimizer is also an ω-minimizer, and the two classes are strictly different. The
interest in ω-minima is motivated by the fact, originally observed in the setting
of Geometric Measure Theory [13, 46, 14], that in many situations minimizers
of constrained problems can be realized as ω-minimizers of unconstrained problems.
Notably, the solutions to obstacle problems and to volume constrained problems are
ω-minima, where the function ω(·) is determined by the properties of the constraint
[18, 92]. The regularity theory for minima extends, sometimes under suitable decay
assumptions of the function ω(·), to ω-minima in a quite satisfying manner. For
instance, Theorem 2.3 extends to any ω-minimizer, as shown in [89, 112], but the
proof is far from being trivial; see also Chapter 7 from [165] for a proof of the
result. In the vectorial case N > 1, assuming that the function ω(·) satisfies (2.15),
partial regularity of minima in the sense of Theorem 4.3 follows; see [92, 99, 207]
for a proof. Therefore the problem of estimating the Hausdorff dimension of the
singular set of ω-minima naturally poses. For the sake of simplicity I will reduce
in the following to discuss simpler functionals of the type (3.6), whereas several
results are available in the case of the complete ones as in (2.1) too. Once again
the problem is the Euler-Lagrange system: even when it exists - and in the case of
F
s
it actually does! - it cannot be used just because ω-minima do not satisfy it,
unless they are real minima. Nevertheless, using the comparison method described
after Theorem 4.8 we have the following result:
Theorem 4.9. Let u ∈ W
1,p
(Ω, R
N
) be an ω-minimizer of the functional F
s
,
under the assumptions (2.14) with µ > 0, and assume that ω(·) satisfies (2.15). If
Σ
u
denotes the singular set of u in the sense of (4.1), then
(4.15) dim
H
(Σ
u
) ≤ n − α , α ∈ (0, 1] .
For the previous result, which improves in certain cases the one in [204], I refer
to the forthcoming paper [208], to which I also refer to for results in the case
of functionals (2.1). Note that estimate (4.15) states that the better ω decays,
with respect to (2.15), the smaller is the singular set, in perfect accordance to the
phenomenon already recorded in Theorems 4.5 and 4.7. Remarkably, and actually
not by chance, estimates (4.15) and (4.13) are the same.
4.6 Boundary problems. I will now briefly report on some recent develop-
ments concerning Dirichlet problems, and partial regularity at the boundary of
solutions to non-linear elliptic systems; these are related to the above singular sets
estimates. Let me consider the following Dirichlet problem associated to the system
(2.8), under the assumptions (2.18):
(4.16)



div a(x, u, Du) = 0 in Ω
u = u
0
on ∂Ω
u
0
∈ C
1,α
(Ω, R
N
) , ∂Ω is C
1,α
-regular .
One can ask if partial regularity carries out up to the boundary. In fact, it is
possible to prove a boundary regularity criterium ensuring that a boundary point
x
0
∈ ∂Ω is regular in the sense that Du is H¨older continuous in neighborhood of
x
0
, of the relative topology of
¯
Ω. Exactly as for (4.3), this is the case if and only if

REGULARITY IN THE DARK SIDE 19
for some small, positive number ε we have
(4.17) −
Z
B(x
0
,R)∩Ω
|Du − (Du)
B(x
0
,R)∩Ω
|
p
dx < ε .
For such a result see [174, 169, 28, 29]. Unfortunately, condition (4.17) does not
yield the existence of regular boundary points, since it is verified a.e. with respect to
the Lebesgue measure, while the boundary ∂Ω is a null set. The problem of finding
the existence of even one regular boundary point remained open, see comments
at page 246 of [152], while, on the other hand, the existence of irregular boundary
points has been known for a while [151], and even for systems with a special, simpler
structure: what a bizarre situation! Moreover, this gap is in sharp contrast to what
happens in the case of elliptic equations, where full regularity carries up to the
boundary [159], and in the case of quasi-linear elliptic systems, i.e. of the form:
div a(x, u)Du = 0, where a.e. boundary point (in the sense of the usual surface
measure) is regular [65, 170, 19, 29]. A first answer to the problem has been given
in [98], building on the work in [248, 249]. The idea is to carry out estimate (4.7) up
to the boundary; then assuming that α is suitably large we have that a.e. boundary
point is regular. We have indeed
Theorem 4.10. Let u ∈ W
1,p
(Ω, R
N
) be a weak solution to (4.16), under the
assumptions (2.18), and assume that
(4.18) α >
1
2
.
Moreover, assume that either n ≤ p + 2, or a(x, u, Du) ≡ a(x, Du). Then almost
every boundary point x ∈ ∂Ω, in the sense of the usual surface measure, is a
regular point, i.e. the gradient is C
0,α
-regular in a neighborhood of x, relative to Ω.
Moreover, when p = 2, we can allow the borderline case α = 1/2.
In fact, under the assumption (4.18), we have that n − 2α < n − 1 and then the
Hausdorff dimension of the boundary singular set is strictly less that the dimension
of the boundary itself; in particular, the existence of regular boundary points fol-
lows. The technique used in [98] is different from that in [248, 249], and rests on a
new, indirect way of treating fractional difference quotients, dealing with them by
a new comparison argument based on convolutions. Such a technique is perhaps
interesting in itself and could find applications elsewhere. It remains the problem
to discuss the existence of regular boundary points when α ∈ (0, 1/2), a case which
is excluded by the methods adopted for Theorem 4.10. A version of Theorem 4.10
valid for minima of a large class of integral functionals, with an integrand which is
convex in the gradient variable, is given in the forthcoming paper [208]. In [208]
we extend to a family of very general functionals the existence results of Jost &
Meier [195], which were valid under a very p eculiar structure assumption on the in-
tegrand F , which was of the type in (4.11) with p = 2; for the case p 6= 2 and again
functionals as in (4.11) see [95]. The boundary ε-regularity criterium for minima
in the sense of (4.17) has been given by Kronz [211], and directly for ω-minima of
quasiconvex integrals. See also the discussion at the end of Paragraph 4.8.
4.7 Conditions for everywhere regularity. A fundamentally important and
open problem is clearly the one of identifying classes of functionals for which ev-
erywhere C
1,α
, or even just continuity, of minimizers, occurs. The same problem
poses for solutions to systems. In other words: are there additional structure as-
sumptions on the integrand F , or on the vector field a, under which the singular
set is void? Up to now, the only known structure preventing the formation of sin-
gularities for minimizers is the one first identified in the fundamental work of K.

20 GIUSEPPE MINGIONE
Uhlenbeck [301]. It prescribes that
(4.19) F (x, v, z) ≡ F (z) = g(|z|) ,
for a suitable function g := [0, ∞] → [0, ∞], such that (2.14) are still satisfied. So,
the dependence of the gradient must occur directly via the modulus |Du|, what
makes, in some sense, the functional “less anisotropic” and rules out singularities
of minima, see [266, 291, 290]. The one in (4.19) is sometimes called “Uhlenbeck
structure” [31]. In the case of systems, the counterpart of (4.19) is
(4.20) a
α
i
(x, v, z) ≡ a
α
i
(z) = h(|z|)z
α
i
α ∈ {1, . . . , N} , i ∈ {1, . . . , n} ,
and h := [0, ∞] → [0, ∞] is once again a suitable function such that (2.18) are
satisfied. An extension to Uhlenbeck’s results can be found in [143, 171], to which
I refer for proofs and references.
The challenge is nowadays to identify new structures, different from the ones
in (4.19)-(4.20), forcing everywhere regularity. In the case of quasi-linear systems
some conditions can be found in [201]. Assuming that F
zz
does not have large
oscillations, it is also possible to prove everywhere regularity: this is a so called
“linearity condition”, see [76] and related references.
4.8 Quasiconvexity. Up to now, I have dealt with convex functionals. Convex-
ity is suitable to ensure lower semicontinuity for variational integrals, and therefore
existence of minima. In the vectorial case there is anyway another condition, much
weaker that convexity, which is sufficient for lower semicontinuity, and actually nec-
essary under certain natural assumptions: this is the so called quasiconvexity. It
makes therefore sense to ask for regularity of minima under such a condition. For
simplicity’s sake, from now on I shall confine myself to consider simpler variational
integrals as in (3.6). A function F : R
nN
→ R is quasiconvex iff
(4.21)
Z
(0,1)
n
[F (z
0
+ Dϕ) − F (z
0
)] dx ≥ 0 ,
for every z
0
∈ R
nN
, and every ϕ ∈ C
∞
((0, 1)
n
, R
N
), with compact support in
(0, 1)
n
. Such a definition deserves comments. First, by a covering argument (0, 1)
n
can be replaced by any other open subset Ω ⊂ R
n
; moreover convex functions are
trivially quasiconvex via Jensen’s inequality, nevertheless the two definitions are
strictly different. Quasiconvexity states that affine functions of the type w
0
+hz
0
, xi,
w
0
, z
0
∈ R
nN
, are minimizers of the functional F
s
in (3.6), in their Dirichlet class. A
large class of quasiconvex functions, strictly intermediate between the one of convex
ones, and the one of quasiconvexes itself, is the class of the so called polyconvex
functions [23, 27, 73, 124]. In the special case u : Ω → R
n
, n = N, these are
integrands g of the form
(4.22)
Z
Ω
g(Dv, Ad Dv, det Dv) dx ,
where g is a convex function of all its arguments, and Ad D v stands for the matrix
of all the minors of Dv.
The difficulty in treating quasiconvex functions largely stems from the non-local
nature of quasiconvexity, as it is immediately clear from definition (4.21). This
point is not fixable: indeed, proving a fundamental and longstanding conjecture of
Morrey, Kristensen [202] showed that there is no local condition characterizing
quasiconvexity. The notion of quasiconvexity was introduced by Morrey [254],
who first identified its connection to lower semicontinuity; the first general lower
semicontinuity result is contained in the seminal paper of Acerbi & Fusco [1], where
it is shown, for instance, that an autonomous quasiconvex functional as in (3.6),
satisfying (2.11), is weakly lower semicontinuous in W
1,p
if and only if F (z) is
quasiconvex. In this last paper the authors also introduced a relevant number

REGULARITY IN THE DARK SIDE 21
of important techniques for treating general lower semicontinuity problems in the
Calculus of Variations. See also [141, 242, 71] for related existence results, while for
further, important progress on lower semicontinuity issues, see [26, 233, 125, 203].
Quasiconvexity plays an important role in the context of non-linear elasticity, as
discussed in the fundamental work of Ball [23]. For further basic information on
quasiconvexity, the reader is referred to [72, 165, 261].
The partial regularity theory for quasiconvex functional was initiated by Evans
[115], who used the following reinforcement of the definition in (4.21):
(4.23) ν
Z
(0,1)
n
(1 + | z
0
|
2
+ |Dϕ|
2
)
p−2
2
|Dϕ|
2
dx ≤
Z
(0,1)
n
[F (z
0
+ Dϕ) − F (z
0
)] dx ,
that he called uniform, strict quasiconvexity, and that serves to provide a sort of
non-degenerate quasi-convexity. For instance, if F is a convex, C
2
-function in the
scalar case N = 1, then (4.23) implies the left hand side inequality in (2.14)
3
for a
different ν > 0, as can be retrieved in [122]. We have the following partial regularity
result:
Theorem 4.11. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F
s
in (3.6), such that F (z) is a C
2
-function satisfying (2.11) and (4.23). Then there
exists an open subset Ω
u
⊂ Ω such that |Ω \ Ω
u
| = 0, and Du ∈ C
0,α
loc
(Ω
u
, R
nN
), for
any α ∈ (0, 1).
This theorem was first obtained by Evans [115], under more restrictive assump-
tions. The version here is due to Acerbi & Fusco [2] in the case p ≥ 2, and to Carozza
& Fusco & myself in the case 1 < p < 2 [56]. Note that for quasiconvex functions
the case 1 < p < 2 cannot b e treated via duality methods starting from the one
p ≥ 2, as in the convex case [171], and it requires a new, direct technical approach.
Again observe that here, unlike in (2.14)
3
, no upper bound on F
zz
is assumed; this
is the main, important contribution of [2], that of course extends also to the convex
case, see for instance [204] for comments. Extensions to complete functionals of
the type in (2.1) are possible, and actually sharply done, while alternative proofs
via the A-approximation method can be found in [92, 96]. A weakening of growth
conditions in a more general case is proposed by Hong [184], while a localization
regularity theorem for minima of functionals which are not necessarily everywhere
quasiconvex has been proved by Acerbi & Fusco [3]. More recently, the partial
regularity of strong local minimizers of quasiconvex functionals has been proved
by Kristensen & Taheri [209]. As far as the lower order regularity is concerned,
there are very few results available, all of them prescribing additional structure
assumptions on the integrand F . For instance assuming that the integrand F is
only asymptotically near (at infinity) to a strongly elliptic quadratic form, Chipot
& Evans [61] proved the Lipschitz continuity of minima; for this and related results
see after Theorem 4.12. Further remarkable extensions, higher integrability results
for minimizers, and even minimizing Young measures, are in the work of Dolzmann
& Kristensen [90].
Note that a partial C
1,α
-regularity theory is also available for certain polyconvex
functionals as in (4.22); for instance, when n = N ≥ 2, the following functional can
be treated:
Z
Ω
¡
|Dv|
p
+ |Ad Dv|
p
+ |det Dv|
p
¢
dx , p > n − 1 .
This time the minimization problem must be settled in appropriate function spaces;
the reader is referred to [144, 145, 114]. An important open problem remains the
one of proving partial regularity of minima of the functional in (4.22) under the

22 GIUSEPPE MINGIONE
“realistic blow-up condition”
(4.24) g ≡ g(z, det z) % ∞ when det z → 0 ,
which has an important meaning in non-linear elasticity; see the very nice review
paper by Ball [24]. The minimizer is this time in the class of preserving orientation
competing maps v i.e. det Dv > 0. An interesting maximum principle in this case
has been recently proved by Leonetti [220]: if the minimization of the functional
in (4.22) is performed in a Dirichlet class with bounded boundary datum, then
the minimizer is itself bounded, provided the rate of blow-up in (4.24) is suitably
controlled, and not too fast. On such a problem, see also some interesting attempts
in [139, 140, 116, 128].
It is important to understand that, due to the non-lo cal character of quasicon-
vexity, the regularity theory for minima of quasiconvex functionals is much more
delicate than that for convex ones in many resp ects. Indeed, roughly speaking,
while convexity allows to compare minimizers with a lot of other maps, quasicon-
vexity restricts the possibility of comparison arguments to lo cally affine-looking
maps only. Moreover, the use of the Euler-Lagrange systems is strictly forbidden!
This clearly appears when looking at critical points of the functional F
s
in (3.6).
Indeed, when considering a convex functional of the simple type (3.6), Theorem
4.3 holds for minimizers, but, via Theorem 4.4, it immediately extends to critical
points, that is solutions to the Euler-Lagrange system (3.7). This is not the case
for quasiconvex functionals. Indeed, even assuming that F is a smooth, uniform
and strict quasiconvex function as in Theorem 4.11, and with p = 2, M¨uller &
ˇ
Sver´ak [262] provided amazing counterexamples of non-minimizing solutions u to
the Euler-Lagrange system (3.7), that are not differentiable on any open subset of
Ω. The problem of determining lower order irregularity of critical points remains
anyway still open, since the solutions exhibited by M¨uller &
ˇ
Sver´ak are still Lip-
schitz continuous. The example of M¨uller &
ˇ
Sver´ak is completely different from
the ones considered in Section 3, and is based on a delicate construction resting
on a the use of the so-called Tartar’s “T
4
-configuration”, and of Gromov’s convex
integration theory [168]; for more on the issue see also their paper with Kirchheim
[200]. The work of M¨uller &
ˇ
Sver´ak is nowadays generating massive developments:
let me mention the remarkable extension of their results to the case of polyconvex
functionals by Sz´ekelyhidi [292], using this time “T
5
-configurations”; see also [293]
for a striking two-dimensional result. Moreover, Bevan [32, 33] has obtained two
dimensional examples of non-C
1
-minimizers of strictly polyconvex functionals, that
is, when the function g appearing in (4.22) is strictly convex. Compare also the
recent work Phillips [268].
At the moment no estimate for the Hausdorff dimension in the general case is
available for minima of quasiconvex functionals, and no analog of Theorem 4.7 has
been proven yet, even for simpler functionals of the type in (3.6), for which estimate
(4.5) is on the other hand available in the convex case. This is essentially due to
the fact that while in the convex case the dimension estimates are obtained by
using the Euler-Lagrange system, and differentiating it in some direct or indirect
way, here, as noted above, such a tool does not yield regularity results in itself.
Under additional assumptions, either on the structure of the integrand F , or on the
regularity of the minimizer u, a few, first results, have been obtained by Kristensen
& myself in [207]. We have indeed the following:
Theorem 4.12. Let u ∈ W
1,∞
(Ω, R
N
) be a local minimizer of the functional F
s
in (3.6), such that F (z) is a C
2
-function satisfying (2.11) and (4.23) with p ≥ 2,
and let Σ
u
:= Ω \ Ω
u
be the singular set of u, in the sense of Theorem 4.11. Then

REGULARITY IN THE DARK SIDE 23
there exists a positive number
(4.25) δ ≡ δ(n, N, p, L/ν, kuk
W
1,∞
) > 0 ,
independent of the minimizer u, such that
(4.26) dim
H
(Σ
u
) ≤ n − δ .
The number δ appearing in (4.25) is in principle explicitly computable by care-
fully keeping track of the constants involved in the proof. It depends on the in-
tegrand F via two features only: first, the modulus of continuity of its second
derivatives F
zz
:
|F
zz
(z
2
) − F
zz
(z
1
)| ≤ γ(|z
1
| + |z
2
|, |z
2
− z
1
|) ∀ z
1
, z
2
∈ R
nN
,
where γ(·, ·) is a non-decreasing, continuous and positive function, such that γ(·, 0) =
0; second: the associated “growth function”
G(M) := sup
|z|≤M
|F
zz
(z)|
1 + |z|
p−2
, M ≥ 0 .
Interestingly, and surprisingly enough, the result of Theorem 4.12 extends to more
general quasiconvex functionals of the the type in (2.1), once again assuming the
H¨older continuity of the function (x, y) 7→ F (x, y, z) in the sense of (2.14)
4
. In
this case, on the contrary of what happened for Theorem 4.7, the number δ is still
independent of the H¨older continuity exponent α in (2.15). On the other hand we
are assuming the minimizer u is already globally Lipschitz continuous. Concerning
such W
1,∞
-assumption, this is verified for a vast class of quasiconvex functionals of
the type (3.6), which are “asymptotically near” the p-Laplacean functional. Indeed,
assume that
(4.27) lim
|z|→+∞
|D
2
F (z) − D
2
H(z)|
|z|
p−2
= 0 ,
where
H(z) := (µ
2
+ |z|
2
)
p/2
, µ ∈ [0, 1] .
A clear model example is given by
(4.28)
Z
Ω
(1 + |Dv|
2
)
p
2
+ g(Dv) dx ,
where g : R
nN
→ R
+
is a C
2
and quasiconvex function (not necessarily strictly),
such that D
2
g(z)/|z|
p−2
→ 0 when z → ∞. For such classes of functionals we
have local Lipschitz continuity of W
1,p
-minimizers, and therefore it follows from
Theorem 4.12 that for every Ω
0
⊂⊂ Ω, there exists a positive number
δ
0
≡ δ
0
(n, N, p, L/ν, dist(Ω
0
, ∂Ω)) > 0 ,
such that dim
H
(Σ
u
∩ Ω
0
) ≤ n − δ
0
. Finally, when assuming (4.27) and minimiz-
ing F
s
in a prescribed Dirichlet class u
0
+ W
1,p
0
(Ω, R
N
), with u
0
∈ W
1,∞
(Ω, R
N
)
and ∂Ω smooth enough, let’s say C
2
, then W
1,p
-minimizers are globally Lipschitz
continuous, and estimate (4.26) works with δ dep ending only on n, N, L/ν, p, ∂Ω
and ku
0
k
W
1,∞
. The interior Lipschitz continuity of minima under the additional
assumption (4.27) has been obtained by Chipot & Evans [61] for p = 2, and ex-
tended to the case p ≥ 2 in [162, 272, 137]. For the global regularity result see [135]
when p = 2, and the recent, interesting paper of Foss [129], when p 6= 2.
Let me observe that Theorem 4.12 extends to ω-minima in the sense of Paragraph
4.5 as well, and therefore improves on the result of Theorem 4.9 when α is small;
see again [207].
The methods of [207] also apply to solutions to the so-called quasimonotone
systems, a notion independently introduced by Fuchs [134], Hamburger [172], and

24 GIUSEPPE MINGIONE
Zhang [306]. These are systems in divergence form of the type in (4.4), where the
ellipticity condition, that is the left hand side inequality in (2.18)
3
, is replaced by
an integral, non-local condition similar to the one in (4.23), that is
ν
Z
(0,1)
n
(1 + |z
0
|
2
+ |Dϕ(x)|
2
)
p−2
2
|Dϕ(x)|
2
dx
≤
Z
(0,1)
n
ha(z
0
+ Dϕ) − a(z
0
), Dϕi dx ,(4.29)
for every z
0
∈ R
nN
, and every ϕ ∈ C
∞
((0, 1)
n
, R
N
) having compact support. Par-
tial regularity of weak solutions to quasimonotone systems holds in the sense of
Theorem 4.11, has been proved by Hamburger [172]. On the other hand, condi-
tion (4.29) is too weak to allow the application of any type of difference quotients
method, and therefore for weak solutions to quasimonotone systems (4.5) cannot
be derived, and no singular set estimate is known. In [207] Theorem 4.12 is seen to
hold for weak solutions to quasimonotone systems too. For regularity and quasi-
monotonicity see also [134, 210], while for existence theorems see [306].
The technique employed for proving Theorem 4.12 is completely different from
the freezing/comparison one used for Theorem 4.7, and in particular no use of
fractional Sobolev spaces is made. On the contrary, we employ certain integral
characterizations of potential spaces in combination with Caccioppoli’s type in-
equalities in order to prove that the singular set Σ
u
enjoys a property known in
Geometric Measure Theory as “set porosity”, see [246]. From this fact estimate
(4.26) follows in a standard way, see also [275] for a wide discussion.
Let me close the paragraph with an interesting open problem, regarding both
quasiconvex functionals and quasimonotone systems. I recommend the reader to
keep in mind here the discussion in Paragraph 4.6, where we have seen that the “ε-
regularity” criterium at the boundary (4.17) also works for minima of quasiconvex
functionals [211]. With some additional efforts this extends also to quasimonotone
systems. As observed in Paragraph 4.6 this does not imply the existence of regular
boundary points. This time Theorem 4.12 does not help: carrying out it up to
the boundary yields no information. Indeed, in general δ is small, while we would
need that n − δ < n − 1 = dim
H
(∂Ω), when ∂Ω is smooth. Even worst, due to
the set-porosity-techniques adopted, the method used in [207] provides a critical
upper bound for δ: δ ≤ 1!! Therefore, in strong contrast to the convex/elliptic
case, to establish the existence of regular boundary points for minima of quasicon-
vex integrals, and solutions to quasimonotone systems, and eventually their almost
everywhere regularity at the boundary, remains an open problem.
4.9 Partial regularity, and degeneration. We have seen that in the vectorial
case N > 1 no degenerate analog of Theorems 4.3 and 4.4 takes place, i.e. we cannot
allow µ = 0, while at the end of the same paragraph a rough explanation of this is
given in terms of the impossibility of linearizing when the gradient of the minimizer
approaches 0. The problem of proving partial regularity for minima of degenerate
(quasiconvex) functionals was raised in [154], Section 3. A first answer has been
given in [101] by Duzaar & myself, where we showed that assuming additional
structure properties on the integrand yields partial regularity in degenerate cases
too. Here I shall restrict, for the sake of simplicity, to the case of functionals of
the type F
s
in (3.6), and rep ort on a special case of the results in [101], that work
directly for quasiconvex integrals. Let me consider a quasiconvex, not necessarily
strictly quasiconvex, C
2
-function g : R
nN
→ R, such that
(4.30) 0 ≤ g(z) ≤ L(1 + |z|
p
) ,

REGULARITY IN THE DARK SIDE 25
and
(4.31) lim
t→0
+
g
z
(tz)
t
p−1
= 0 ,
uniformly on the set {z ∈ R
nN
: |z| = 1}. Now, let me define the following
degenerate quasiconvex functional:
(4.32) DQ(v) :=
Z
Ω
ν|Dv|
p
+ g(Dv) dx ,
with ν > 0. Then we have
Theorem 4.13. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional DQ,
under the assumptions (4.31), and (4.32). Then there exists an open subset Ω
u
⊂ Ω
such that |Ω \ Ω
u
| = 0, and Du ∈ C
1,α
loc
(Ω
u
, R
n
) for some α ∈ (0, 1).
For a more general result see again [101]. The proof of Theorem 4.13 offers
an example of application of the “p-harmonic approximation lemma”, obtained in
[100], that extends the original De Giorgi’s one [78, 283] to the case p 6= 2. I feel
that in the context of the p-Laplacean theory, and in the one of harmonic mappings,
it has its own interest. I therefore find it worth reporting the statement here.
Theorem 4.14. (p-harmonic approximation lemma). Let B be the unit ball in
R
n
. For each ε > 0 there exists a positive constant δ ∈ (0, 1] depending only on
n, N, p, and ε, such that the following is true: Whenever u ∈ W
1,p
(B, R
N
) with
R
B
|Du|
p
dx ≤ 1 is approximately p-harmonic in the sense that
¯
¯
¯
¯
Z
B
|Du|
p−2
Du · Dϕ dx
¯
¯
¯
¯
≤ δ sup
B
%
|Dϕ|
holds for all ϕ ∈ C
1
c
(B, R
N
), there exists a p-harmonic function h ∈ W
1,p
(B, R
N
)
such that
Z
B
|Dh|
p
dx ≤ 1 and
Z
B
|h − u|
p
dx ≤ ε
p
.
A function h is of course said to be p-harmonic when div (|Dh|
p−2
Dh) = 0. The
idea for proving Theorem 4.13, inspired by the original strategy of Uhlenb eck [301],
is that when the gradient Du of the minimizer is far from 0 in a quantitatively
determined way, then one can linearize and prove partial regularity as discussed
in Paragraph 4.2. At this stage the linearization is achieved via the A-harmonic
approximation method of Duzaar & Steffen [104], which is a variant of Theorem
4.14 for p = 2. On the contrary, when the gradient is near zero, so that the problem
becomes really degenerate, one directly compares u with minimizers of the func-
tional (2.16), which are regular by the results in [301], compare Paragraph 4.7, and
good regularity estimates for Du still follow. This time the comparison argument
is achieved via Theorem 4.14, which is useful when treating truly degenerate situa-
tions. This last step is possible since (4.32) tells us that near z = 0, the integrand of
DQ b ehaves essentially as ν|z|
p
. Finally one shows that the classical harmonic (or
A-harmonic) approximation lemma, and the new p-harmonic one, perfectly match
in a suitable iteration procedure.
As a final observation let me mention that when p = 2 the parabolic analogue of
Theorem 4.14, and therefore of the original De Giorgi’s lemma, has been obtained
in [102]; for the case p 6= 2 proving such a parabolic analogue remains an open
problem.

26 GIUSEPPE MINGIONE
5. Irregularity strikes back
In this section, and in the following one, I shall restrict my attention to func-
tionals of the type
(5.1) F(v) :=
Z
Ω
F (x, Dv) dx .
We have seen that, while in the vectorial case N > 1 there is no hope to get
everywhere regularity of minimizers for general variational integrals as in (2.1), at
least in the scalar one N = 1, everywhere regularity in the interior of Ω is guaranteed
under reasonable assumptions: this is essentially the content of Section 2. All the
results in Section 2 follow assuming at least one common, main condition, that is
(2.11). Now let’s give a look at the following functionals, where 1 < p < q are fixed
numbers:
F
1
(v, Ω) =
Z
Ω
|Dv|
p
log(1 + |Dv|) dx ;
F
2
(v, Ω) =
Z
Ω
n
X
i=1
a
i
(x)|D
i
v|
p
i
dx ,
1 ≤ a
i
(x) ≤ L, 1 < p := p
1
≤ p
2
≤ . . . ≤ p
n
=: q ;
F
3
(v, Ω) =
Z
Ω
|Dv|
p
+ a(x)|Dv|
q
dx, 0 ≤ a(x) ≤ L ;
F
4
(v, Ω) ≡ D
p(x)
(v) =
Z
Ω
|Dv|
p(x)
dx, 1 < p ≤ p(x) ≤ q ;
F
5
(v, Ω) =
Z
Ω
|Dv|
p(x)B(|Dv|)
dx, 1 < p ≤ p(x) ≤ q − γ, 1 ≤ B(|z|) ≤ γ ;
F
6
(v, Ω) =
Z
Ω
|Dv|
p(x)
B(|Dv|) dx,
1 < p ≤ p(x) ≤ q − q
1
, 1 ≤ B(|z|) ≤ L(1 + |z|
q
1
) .
None of the integrands corresponding to the functionals F
1
−F
6
satisfies conditions
(2.11), for any possible choice of the exponent p ≥ 1. But all of them satisfy, for
the correspondingly specified choice of the numbers (p, q), and suitable ν, and L,
the new, more general growth conditions
(5.2) ν|z|
p
− L ≤ F (x, z) ≤ L(1 + |z|
q
) , 1 < p < q .
Functionals satisfying conditions (5.2), and not meeting the ones in (2.11), are called
functionals with (p, q)-growth conditions. Here I am following the terminology of
Marcellini, who was the first to initiate a systematic study of such integrals in a
series of seminal papers [235, 236, 237, 238, 239]. A related notion can be given for
equations, but I am not going to deal with them in this paper. Before going on,
let me point out some permanent assumptions. Due to the (p, q)-growth conditions
satisfied by the integrand F , the following more general definition of minimality is
usually adopted in this case: A map u ∈ W
1,1
loc
(Ω, R
N
) is a local minimizer of F iff
x → F (x, Du(x)) ∈ L
1
loc
(Ω) and
Z
supp ϕ
F (x, Du) dx ≤
Z
supp ϕ
F (x, Du + Dϕ) dx ,
for any ϕ ∈ W
1,1
(Ω, R
N
) such that supp ϕ ⊂ Ω. From this definition it immedi-
ately follows that any local minimizer is in W
1,p
loc
(Ω, R
N
), since the left hand side
inequality in (5.2). On the other hand, the same left hand side inequality in (5.2)
guarantees coercivity of F in W
1,p
(Ω, R
N
), and therefore, provided the functional
itself is lower semicontinuous in the weak topology of W
1,p
loc
(Ω, R
N
), Direct Methods
of the Calculus of Variations guarantee the existence of minimizers in W
1,p
(Ω, R
N
)

REGULARITY IN THE DARK SIDE 27
with fixed boundary data. Lower semicontinuity under (p, q) growth conditions can
be achieved by just imposing, for instance, that F is convex in the gradient variable,
see [165], Chapter 4.
In order to show the impact of (p, q)-growth conditions on the regularity and/or
irregularity of minima, I will start with examples of Marcellini, which elaborate
upon previous counterexamples by Marcellini himself [234], and Giaquinta [155].
5.1 A first type of examples (Marcellini [236]). Marcellini considered a family
of elliptic equations and integral functionals satisfying (p, q)-growth conditions, a
particular case of which being
M(v) :=
Z
Ω
n−1
X
i=1
|D
i
v|
2
+
1
2
|D
n
v|
4
dx ,
and proved the existence of unbounded solutions provided
(5.3) q >
(n − 1)p
n − 1 − p
, n > 2 , 1 < p < n − 1 .
For instance, when n ≥ 6 the unbounded function
u(x) :=
r
n − 4
24
x
2
n
q
P
n−1
i=1
x
2
i
,
is the unique minimizer of M in its Dirichlet class.
It is interesting to see that Marcellini’s examples are concerned with degenerate
integrals. For instance, when |D
n
u| approaches 0 the Euler-Lagrange equation of
the functional M
n−1
X
i=1
D
ii
u +
1
2
D
n
(|D
n
u|
2
D
n
u) = 0 ,
becomes degenerate elliptic, loosing ellipticity in the x
n
-direction. This point was
fixed by Hong [185], who considered the regular, non-degenerate elliptic functional
H(v) :=
Z
Ω
|Dv|
2
+
1
2
|D
n
v|
4
dx ,
having a regular integrand, and exhibiting the following minimizer for n ≥ 6:
u(x) :=
r
n − 4
24
x
2
n
q
P
n−1
i=1
x
2
i
−
2
n − 2
r
n − 4
24
v
u
u
t
n−1
X
i=1
x
2
i
.
Hong’s example is useful because it confirms the intuitive fact that for functionals
with (p, q)-growth, problems mainly come for the behavior of the integrand F (x, z)
for large values of |z|. For a further discussion on counterexamples see the survey
of Leonetti [219].
5.2 A second type of examples (Fonseca & Mal´y & myself [126]). These work
for non-autonomous functionals of the type (5.1). A one-point, isolated singularity
version of them has been previously obtained in [111], elaborating on some construc-
tions of Zhikov [308] in the theory of Lavrentiev Phenomenon, see also Paragraph
6.5 below. The examples show that, provided p and q are far enough, depending on
the dimension n, and the regularity of x 7→ F (x, z), then the set of non-Lebesgue
points of minimizers can be nearly as bad as any that of any other W
1,p
-function.
Indeed, a well known measure theoretic result states that the set of non-Leb esgue
points of (the precise representative of) a W
1,p
-function has Hausdorff dimension
not larger than the maximal dimension n − p, see for instance [165], Chapter 2.
Here it is possible to find a minimizer of a convex, regular, and scalar variational

28 GIUSEPPE MINGIONE
integral, whose set of non-Lebesgue points is a Cantor-type set of (nearly) maximal
dimension. Indeed, we have
Theorem 5.1. For every choice of the parameters
(5.4) 2 ≤ n , α ∈ (0, ∞) , 1 < p < n < n + α < q < ∞ , ε > 0 ,
there exist a functional
(5.5) F
3.2
: u 7→
Z
Ω
£
(1 + |Du|
2
)
p
2
+ a(x)(1 + |Du|
2
)
q
2
¤
dx , u ∈ W
1,p
(Ω) ,
with Ω ⊂ R
N
being a bounded Lipschitz domain, a ∈ C
α
(Ω), a ≥ 0, a local mini-
mizer u ∈ W
1,p
(Ω) of F, and a closed set Σ ⊂ Ω with
dim
H
(Σ) > n − p − ε ,
such that all the points of Σ are non-Lebesgue points of (the precise representative
of) u.
As it will b e clear from Paragraph 6.5, and in particular from Theorem 6.6,
the condition on the distance between p and q in previous theorem cannot be
relaxed. In the previous example the set of non-Lebesgue points of the minimizer
is unrectifiable. This is the effect of the presence of x in the integrand, allowing to
“distribute” the singularities of the minimizer on a Cantor type set. More comments
can be found in Paragraph 6.5.
6. The return of regularity: (p, q)-growth conditions
After some sporadic come out in the literature, see for instance [303] and re-
lated references, the study of regularity of minima of functionals with non-standard
growth of (p, q)-type has been initiated by Marcellini [235, 236, 237, 238, 239],
who first identified a condition that, under suitable smoothness assumptions on
the integrand F , ensures the regularity of minima. When referring to (p, q)-growth
conditions (5.2), let me call the quantity q/p > 1, the gap ratio of the integrand F ,
or simply, the gap. Marcellini’s approach prescribes that
(6.1) “the gap
q
p
cannot differ too much from 1” ,
in other words the numbers q and p cannot be too far apart. This approach is of
course suggested by the counterexamples we have seen in Section 5, and in particular
by (5.3) and (5.4). The application of (6.1) will be a standard in the next theorems,
and the choice of the bound to assume on the gap q/p will change accordingly to
the specific situation.
6.1 Lipschitz regularity and the gap. In order to give a first instance of the
effect of assuming (6.1), I will present two sample theorems, which are not the most
general ones available in the literature, but that are nevertheless suitable to give a
correct kind of flavor of the matter. I shall start considering simpler, autonomous
functionals of the type
(6.2)
F
s
(
v
) :=
Z
Ω
F
(
Dv
)
dx ,
with the integrand F (z) satisfying the following “(p, q)-version” of assumptions
(2.14):
(6.3)











z 7→ F (z) is C
2
ν|z|
p
≤ F (z) ≤ L(1 + |z|
q
)
ν(1 + |z|
2
)
p−2
2
|λ|
2
≤ hF
zz
(z)λ, λi ≤ L(1 + |z|
2
)
q−2
2
|λ|
2
.

REGULARITY IN THE DARK SIDE 29
Note that at this point the convexity assumptions (6.3)
3
are formulated according
to the growth conditions in (6.3)
2
. Otherwise differently specified, I am dealing
with the general vectorial case u : Ω → R
N
, N ≥ 1. The following scalar result of
Marcellini is taken from [236]:
Theorem 6.1. Let u ∈ W
1,q
loc
(Ω) be a local minimizer of the functional F
s
, under
the assumptions (6.3), in the scalar case N = 1; moreover, assume that
(6.4)
q
p
<
n
n − 2
when n > 2 .
Then Du ∈ L
∞
loc
(Ω, R
n
).
Here we see that assuming (6.1) in the form of (6.4) allows to get the local
boundedness of the gradient. In most cases this is the focal point of regularity
for functionals with (p, q)-growth. Indeed, once this kind of result is achieved,
for simpler functionals of the type in (6.2) the higher regularity problem can be
dealt with as for functionals with standard growth conditions (2.11). Roughly
speaking, unless we are not dealing with degenerate problems, the behavior of a
non-standard growth functional differs from that of the standard ones only for the
growth conditions in the gradient variable z, and therefore for large values of z. So,
when already knowing that the minimizer u has a bounded gradient, the behavior
at infinity of the function F becomes irrelevant, and the standard, higher regularity
theory applies. This argumentation can be of course made rigorous, see for instance
[238, 251].
Theorem 6.1 cannot extend to the vectorial case N > 1, as it is clear from the
counterexamples valid already when p = q, see Section 3. For Theorem 6.1 one
assumes that minimizers are a priori in u ∈ W
1,q
loc
(Ω), while in general we have seen
that they are a priori only in u ∈ W
1,p
loc
(Ω), compare previous section. Getting rid of
this integrability gap is actually the first step when proving regularity of minimizers
under (p, q)-growth conditions: passing from u ∈ W
1,p
loc
to u ∈ W
1,q
loc
. This is crucial
when proving higher regularity such as Lipschitz continuity, since by the right side
inequality (6.3)
2
, the integral of |Du|
q
appears everywhere in the estimates. The
following result is due to Esposito & Leonetti & myself, and is a particular case of
the ones in [109]:
Theorem 6.2. Let u ∈ W
1,p
loc
(Ω, R
N
) be a local minimizer of the functional F
s
,
under the assumptions (6.3); moreover, assume that
(6.5)
q
p
<
n + 2
n
.
Then u ∈ W
1,q
loc
(Ω, R
N
).
By combining Theorems 6.1 and 6.2 we obtain the following:
Theorem 6.3. Let u ∈ W
1,p
loc
(Ω) be a local minimizer of the functional F
s
, under
the assumptions (6.3), in the scalar case N = 1; moreover, assume that the gap
q/p satisfies (6.5). Then D u ∈ L
∞
loc
(Ω, R
n
).
For a discussion on the borderline case q = p(n + 2)/n see also [103]. Further
extensions are in [110]. Assuming the “Uhlenbeck structure”, see Paragraph 4.7,
has effects also in the case of functionals with non-standard growth, see [238], and
again [110] with [251]. Indeed we have
Theorem 6.4. Let u ∈ W
1,p
loc
(Ω, R
N
) be a local minimizer of the functional F
s
,
under the assumptions (6.3); moreover, assume that the gap q/p satisfies (6.5), and
that the integrand F can be written as F (z) ≡ g(|z|). Then Du ∈ L
∞
loc
(Ω, R
nN
).

30 GIUSEPPE MINGIONE
The above results are concerned either with higher integrability of Du, or with
its Lipschitz continuity. In the vectorial case a Partial regularity theory is also
available, though not being yet as complete as one would hop e. I refer to the papers
[138, 113, 38, 39, 40], for partial regularity results under (p, q)-growth conditions.
6.2 On the gap condition (6.1). The reader will immediately notice that the
bound in (6.4) is larger than the one in (6.5). Actually, except in a few special
cases, it is not known in general neither if it possible to assume (6.4) in Theorem
6.2, nor what is the best bound for the gap q/p one has to assume when applying
the principle in (6.1) to autonomous functionals of the type in (6.2): this is a main
open problem in the theory. Anyway, observe that both the bound in (6.2), and the
one in (6.5), are smaller that the one allowing for the counterexample to regularity
in (5.3). The situation changes when considering non-autonomous functionals of
the typ e in (5.1), for which the best bound for q/p is known, compare Paragraph
6.5 below.
Before going on, I shall try to give a very rough explanation of why (6.1) comes
up naturally as a condition ensuring regularity. Apart from those ones obviously
given by the counterexamples of Section 5, there are indeed also technical reasons
for considering (6.1) as a natural assumption. We have seen in Paragraph 2.1 that
the regularity of solutions to linear elliptic equations as in (2.4) strongly depends on
how large the ellipticity ratio L/ν is; compare for instance (2.6), or the singular sets
estimates of Theorems 4.5-4.6. Now, modulo a suitable approximation argument,
Theorems 6.1-6.4 are proved making use of the Euler-Lagrange equation (when in
the scalar case, to which I restrict for the following discussion) of the functional F:
Z
Ω
n
X
i=1
F
z
i
(Du)D
i
ϕ dx = 0 , ∀ ϕ ∈ C
∞
c
(Ω) .
The following argumentation will be now purely formal, but it can be raised to the
correct standard of rigor. In the previous equation let me take D
s
ϕ, instead of ϕ;
then let me integrate by parts
Z
Ω
n
X
i,j=1
F
z
i
z
j
(Du)D
j
(D
s
u)D
i
ϕ dx = 0 .
Therefore, letting a
i,j
:= F
z
i
z
j
(Du(x)), the function w := D
s
u satisfies the following
linear elliptic equation with measurable coefficients
(6.6)
Z
Ω
a
i,j
(x)D
j
wD
i
ϕ dx = 0 , ∀ ϕ ∈ C
∞
c
(Ω) ,
which is of the type (2.4), but this time the co efficients are elliptic but not bounded
(6.7) |a
i,j
(x)| ≤ L(1 + |Du(x)|
2
)
q−2
2
, a
i,j
(x)λ
i
λ
j
≥ ν(1 + |Du(x)|
2
)
p−2
2
|λ|
2
.
The ellipticity ratio of the matrix {a
i,j
(x)}, that is the ratio between the largest
and the lowest eigenvalue of {a
i,j
(x)}, can be this time bounded only by
(6.8) R(Du) := L/ν(1 + |Du(x)|
2
)
q−p
2
,
which blows-up when |Du| → ∞. This tells us that (6.6) is an instance of a non-
uniformly elliptic equation, an equation where the ratio between the largest and the
lowest eigenvalue is not assumed to be a-priori bounded, and it actually depends
on the solution itself. We have therefore a very critical situation: we would like
to prove that the gradient is bounded, but on the other hand the ellipticity ratio,
which is the quantity controlling the regularity of solutions, blows-up exactly when
|Du| → ∞. At this stage the role of assuming a condition like (6.1) becomes clear:
it serves to give a bound to the rate of possible blow-up of R(Du). In other words,
(6.1) controls the rate of non-uniform ellipticity of equation (6.6): if R(Du) does

REGULARITY IN THE DARK SIDE 31
not potentially blow-up very fast in terms of Du, and this happens when q/p is
not very large, then it actually stays bounded, otherwise it really blows-up! A
precursor of gradient bounds for general non-uniformly elliptic equation is Simon,
in his beautiful paper [281]. Here, Simon’s conditions are a sort of re-formulation of
the principle (6.1). Again, on non-uniformly elliptic equations see [214, 298, 280].
There are anyway cases where condition (6.1) can be assumed in a form weaker
than the ones considered up to now. In particular we have seen that Theorems
6.1-6.4 require that q/p → 1 when n → ∞. By looking at the counterexamples
in Paragraph 5.1, we see that if in general q/p 6→ 1, then minimizers become
unbounded. Now it happens that when minimizers are bounded, or their boundary
data is assumed to be bounded and the functional allows for a maximum principle,
then we do not have to require that q/p → 1 when n → ∞ in order to prove
regularity results for the gradient. We can just assume, for instance, q < p + 2; this
is essentially the strategy introduced in [108], and then developed in the papers
[37, 39]. See also the interesting maximum principle papers [145, 216].
6.3 Additional structures. Concerning functionals with (p, q)-growth condi-
tions, there has been a large literature over the last fifteen years. Let me emphasize
a few main directions. A first one concerns the so called anisotropic functionals,
whose model is given in Section 5 by the functional F
2
. In this case the functional
satisfies an additional structure/growth condition of the type
(6.9)
n
X
i=1
|z
i
|
p
i
≤ F (z) ≤ L(1 +
n
X
i=1
|z
i
|
p
i
) .
The notation here is as for functional F
2
in the previous section. The essence
here is that every directional derivative is penalized with its own exponent; these
functionals are naturally defined in the so called anisotropic Sobolev spaces [4], and
more precise results can be obtained thanks to the peculiar structure coming into
the play, and yielding more information. Papers dedicated to the issue are, amongst
the others, [303, 235, 147, 149, 35, 217, 218, 4, 43, 289, 225, 63, 226]. Assuming an
additional structure as in (6.9), that is assuming that the integrand F (z) is bounded
from above and below by the same quantity, leads to better results in other cases.
A relevant one is that of functionals naturally defined in Orlicz spaces, that is when
we have growth and coercivity conditions of the type
(6.10) Φ(|z|) ≤ F (x, v, z) ≤ L(1 + Φ(|z|)) .
Here Φ : [0, ∞] → [0, ∞] is a Young function i.e: a convex, increasing function such
that Φ(0) = 0; see [271] for a comprehensive introduction to Orlicz spaces and more
information on Young functions. If in turn Φ(t) satisfies νt
p
−L ≤ Φ(t) ≤ L(1+ t
q
),
then the function F (z) also exhibits (p, q)-growth conditions. There is a large
literature dedicated to functionals satisfying (6.10). In many of these papers the
crucial assumption on the function Φ is the so called 4
2
-condition
(6.11) Φ(2t) ≤ cΦ(t) ,
that serves to exclude fast growth instances such as Φ(t) ≡ exp(t
2
). Moreover,
another condition, namely the ∇
2
-condition is also imposed, see [74], a condition
dual to the one in (6.11): it serves to exclude slow growth instances such as Φ(t) ≡
t log(1 + t
2
). On such cases I will nevertheless turn back in the next paragraph.
The basic, common approach in the papers dedicated to the structure (6.10) is
to reproduce the results valid for functionals satisfying (2.11), viewing the case
Φ(t) = t
p
as a special one of (6.10). In a certain sense the function F (z) is now
“re-balanced” by assuming (6.10). Papers dedicated to the issue are, amongst
the others, [34, 294, 148, 62, 257, 146, 244, 245, 74]. I hereby want to mention
that an interesting bridge between (6.9) and (6.10) has been built by Cianchi [63],

32 GIUSEPPE MINGIONE
while a very complete picture concerning also certain classes of elliptic equations
in divergence form is given by Lieberman [223], relying on the techniques in [281].
6.4 Extreme cases. Considering (p, q)-growth conditions turns out to be still
too restrictive when dealing with certain classes of variational integrals. Here are
two examples of functionals not satisfying (p, q)-growth conditions, and for opposite
reasons:
F
7
(u) =
Z
Ω
|Du| log(1 + |D u |) dx, F
8
(u) :=
Z
Ω
exp(|Du|
2
) dx .
The former one does not meet ( p, q)-growth conditions because the integrand grows
too slowly in the gradient variable, and it fails to be polynomially super-linear; the
latter because the integrand grows faster than any power. Nevertheless for both of
the functionals a fully satisfying regularity theory is available, even in the vectorial
case N ≥ 1 (this is also due to the “Uhlenbeck structure” shared by both the
integrands, see Paragraph 4.7).
As for F
8
, the C
∞
-nature of minimizers was shown by Lieberman [224], who
relied very much on the peculiar structure of the integrand. A much more general
theory is offered by Marcellini [238, 239], who is able to treat, also in the case
N ≥ 1, a very wide class of variational integrals with fast growth in the gradient,
including any finite composition of exponentials i.e. functionals of the form
(6.12)
Z
Ω
exp(exp(. . . exp(|Du|
2
) . . .)) dx .
Extensions to a class of non-autonomous integrals can be found in [243].
As for F
7
, the continuity of the gradient was obtained in two dimensions n = 2
by Frehse & Seregin [133] and Fuchs & Seregin [138], who explicitly raised the
problem of proving the result in higher dimensions. This was settled by Siepe &
myself in [251], where the proof of the H¨older continuity of the gradient of minima
in any dimension n > 2 is achieved in the general vectorial case N ≥ 1. The proof
in [251] relies on the simple observation that, no matter how slow the integrand
F (z) = |z| log(1 + |z|) grows, when looking at the the second derivative matrix F
zz
,
it does not decay fast enough yet to allow for irregularity. In other words, there is
ellipticity enough to regularize minimizers. The same observation allows to treat
more general integrals with slower growth, and for instance any finite composition
of logarithms is allowed, i.e. functionals growing like
(6.13)
Z
Ω
|Du| log(1 + log(. . . log(1 + |Du|
2
) . . .)) dx .
This last one is in some sense “dual” to that in (6.12), in the same way F
7
and
F
8
can be considered dual each other (there’s a way to make this rigorous, using
the theory of duality in Orlicz spaces, see [271]). The regularity for minima of
functionals as in (6.13) has been obtained by Fuchs & myself in [136]. Further
developments can be found in [36, 241].
6.5 Non-autonomous functionals. Up to now I confined myself to simple,
autonomous integrands of the type in (6.2). Now I am going to deal with more
general, non-autonomous ones of the type
(6.14) F
na
(v) :=
Z
Ω
F (x, Dv) dx ,
still satisfying non-standard growth conditions of (p, q)-type (5.2). If we look at
the the case p = q, and especially at Theorem 2.3 and at Paragraph 2.2, we see
that the precise degree of regularity of the integrand F (x, z) with respect to the x-
variable is irrelevant in order to get the H¨older continuity of minimizers. Moreover,
also when looking at Theorem 2.5, wee see that the degree of H¨older continuity

REGULARITY IN THE DARK SIDE 33
of F (x, z) with respect to x only influences the degree of H¨older continuity of
Du, but not the fact that Du is H¨older continuous or not; in other words, any
degree α of H¨older continuity of x → F (x, z), suffices in order to get a H¨older
continuous gradient. The modest influence of the presence of the x-variable in the
integrand is also clear when looking at the techniques of proof of Theorems 2.4
and 2.5, see [68, 158, 231], where the presence of x is treated essentially using local
perturbation methods. When dealing with (p, q)-growth conditions the situation
drastically changes, and the novelty is that the presence of x cannot be treated as
a perturbation anymore. This can be guessed by looking at the structure of the
integrands in functionals F
3
, F
4
in Section 5, that I will call F
3
(x, z) and F
4
(x, z),
respectively. In both cases, if we keep x fixed and let z vary, the integrand exhibits
standard growth conditions; for instance z → F
3
(x, z) has p-growth if a(x) = 0,
and it has q-growth if a(x), while it globally exhibits (p, q)-growth conditions since
the variable x is varying simultaneously with z. A similar argumentation works
of course for the integrand of F
3.2
in Paragraph 5.2, and for F
4
(x, z) of F
4
, with
which the next section is concerned. This immediately tells us that in the case of
functionals with (p, q)-growth, the effect of x can be very relevant, since it is itself
responsible for the (p, q)-growth conditions to appear! Indeed, as demonstrated
in the papers [111, 126], when dealing with functionals of the type (6.14), the
regularity of minima is ruled by a subtle interaction between the the regularity of
the function x → F (x, z), and the size of the gap q/p. The counterexample of
Paragraph 5.2 already tells us that, in order to create singularities, the numbers q
and p must be far accordingly to the size of α. This is a general phenomenon, that
reveals to be another instance of principle (6.1): for functionals of the type (6.14),
the condition allowing to prove regularity results for minimizers is
(6.15)
q
p
<
n + α
n
, α ∈ (0, 1] ,
where this time α is the H¨older continuity exponent of x → F (x, z)/(1 + |z|
q
),
compare (6.19)
4
below. Therefore: the less regular with respect to x the integrand
F (x, z) is, the less we are allowed to get q/p far from 1. Condition (6.15) is actually
sharp, as the counterexample from Paragraph 5.2 immediately shows. For instance,
let me report the following result, taken from [111]:
Theorem 6.5. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F
3
,
and assume that 0 ≤ a ∈ C
0,α
(Ω), with (6.15). Then Du ∈ W
1,q
loc
(Ω, R
nN
).
There is even a more precise version, arising when minimizers are bounded:
in this case condition (6.15) can be replaced by q ≤ p + α, which is again the
one appearing in Theorem 5.1. This is according to the discussion at the end of
Paragraph 6.2; see [103] for more details, and also for the borderline cases q =
p(n + α)/n and p = q + α, when previous Theorem 6.5 is still valid. (I hope this
paper will be ready soon!)
Let me notice that there is clearly a gap between the autonomous condition (6.5),
and the non-autonomous one (6.15). Even in the most favorable case α = 1 the two
conditions do not coincide, and (6.15) is more restrictive than (6.5), this being a
non-fixable effect of the presence of x. On the other hand, a pleasant consequence of
Theorem 6.5, and of Theorem 6.6 below, is that while for autonomous functionals
of the type (3.6) it is not known what is in general the best bound to assume
on the gap q/p, compare Paragraph 6.2, here we see that in the non-autonomous
case the best possible bound is the one (6.15). This is again a consequence of the
counterexample in Paragraph 5.2.
Theorem 6.6 is a particular case of a more general theory, whose beginnings
are settled down in [111], and that I am now going to outline. This goes via the

34 GIUSEPPE MINGIONE
analysis of the so-called Lavrentiev Phenomenon (LP), that functionals of the type
in (6.14) typically exhibit when under (p, q)-growth conditions. Roughly speaking,
LP occurs at a map v ∈ W
1,p
(Ω, R
N
), when it is not possible to find a sequence of
more regular maps v
n
∈ W
1,q
loc
(Ω, R
N
), such that v
n
* v in W
1,p
loc
(Ω, R
N
), and the
following “approximation in energy” takes place:
(6.16)
Z
A
F (x, Dv
n
) dx →
Z
A
F (x, Dv) dx ,
for every A ⊂⊂ Ω, with A being an open subset. This is actually a re-adaptation
of the original definition that perfectly fits here, and that I am adopting from now
on for the sake of simplicity. When the Lavrentiev Phenomenon occurs at a local
minimizer u it then follows, in particular, that it is not possible to realize locally
minimizing sequences {u
n
}
n
for F, with more regular maps u
n
∈ W
1,q
loc
(Ω, R
N
).
LP is a clear obstruction to minimality, since if u is a minimizer such that u ∈
W
1,q
loc
(Ω, R
N
), then by the very definition there is not LP at u. It is interesting, and
significant, to see that F never exhibits LP either when p = q, or when F (x, z) ≡
F (z), see [111], so that, LP results from the coupling of (p, q)-growth conditions with
dependence on x in the integrand. For a nice survey on LP, I recommend [48], while,
in the setting of functionals with (p, q)-growth, fundamental contributions are due
to Zhikov [308, 309], where several examples of LP are given. A striking example
of LP for functionals with variable growth exponent (see the last section) has been
offered by Foss [127]; LP also play an important role in non-linear elasticity, see
[25, 130, 131]. A useful way to quantify LP can be introduced according to Buttazzo
& Mizel [49], as I will briefly explain now. From now I shall consider integrands
F (x, z) which are convex with respect to z, in order to gain lower semicontinuity
for the related functionals; a more general situation can be found again in [49, 111].
Following [49], let me define the following relaxed functional
¯
F(v, B
r
) := inf
v
n
½
lim inf
n
Z
B
r
F (x, Dv
n
) dx : v
n
∈ W
1,q
(B
r
, R
N
),
v
n
* v in W
1,p
(B
r
, R
N
)
ª
,(6.17)
where B
r
⊂⊂ Ω is a ball with radius r > 0. As mentioned above, since F (x, z) is
convex with respect to z, then
F
na
(v, B
r
) :=
Z
B
r
F (x, Dv) dx ≤ lim inf
n
Z
B
r
F (x, Dv
n
) dx ,
whenever v
n
* v, and v
n
∈ W
1,p
(B
r
, R
N
). Therefore F
na
(v, B
r
) ≤
¯
F(v, B
r
) for
every v ∈ W
1,p
(B
r
, R
N
), and it is possible to define the following, non-negative
Lavrentiev Gap Functional:
G(v, B
r
) :=
¯
F(v, B
r
) − F
na
(v, B
r
) ≥ 0 , ∀ v ∈ W
1,p
(B
r
, R
N
) .
The value of the functional G(v, B
r
) gives a measure of the impossibility of ap-
proximating in energy, that is (6.16), the map v with a sequence of more regular
maps. Indeed (6.16) occurs with A ≡ B
r
, if and only if G(v, B
r
) = 0. Now, let
u ∈ W
1,p
loc
(Ω, R
N
) be a local minimizer of F
na
, then, clearly
(6.18) u ∈ W
1,q
loc
(Ω, R
N
) =⇒ G(u, B
r
) = 0 , ∀ B
r
⊂⊂ Ω .
This leads to the following developments: usually one proves regularity of minimiz-
ers to exclude the LP, that is (6.18); in [111] we followed the opposite procedure,

REGULARITY IN THE DARK SIDE 35
using the absence of LP to prove regularity. Let me consider the following assump-
tions, parallel to those in (2.14):
(6.19)



















z 7→ F (x, z) is C
2
ν|z|
p
≤ F (x, z) ≤ L(1 + |z|
q
)
ν|z|
p−2
|λ|
2
≤ hF
zz
(x, z)λ, λi ≤ L(1 + |z|
2
)
q−2
2
|λ|
2
|F (x, z) − F (y, z)| ≤ L|x − y|
α
(1 + |z|
q
) .
Then we have the following result from [111], with a few improvements from [103]:
Theorem 6.6. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional F
na
,
under the assumptions (6.19). Assume also that (6.15) holds, together with
(6.20) G(u, B
r
) = 0 .
Then Du ∈ W
1,q
loc
(Ω, R
nN
).
Condition (6.20) is not tautological, and it is necessary by (6.18), but it may
appear difficult to verify in general. Nevertheless, a twist happ ens now: for all the
model cases in the literature, and in particular for the functionals F
1
− F
6
, we have
(6.21) (6.15) =⇒ (6.20) ,
establishing a deeper connection between the regularity theory and LP. Therefore,
for all the model cases known, and in particular for F
1
− F
6
from Section 5, as-
sumptions (6.19) and (6.15) suffice for the basic W
1,q
regularity of minima. This
path is again followed in [111], building on previous work of Zhikov, where large
classes for functionals for which (6.21) occurs have been demonstrated. Further
results on LP can be found in [250]. It remains an open problem to establish the
validity of (6.21), under the only assumptions (6.19). Interesting developments to
[111] can be found in [30, 40, 57, 69].
7. Variable growth exponents
We have seen that in the case of functionals with (p, q)-growth of the type (5.1),
the regularity theory for minima is still far from finding a definitive and general
setting, being linked, for instance, to the analysis of Lavrentiev Phenomenon. I have
to say that my feeling is that, in the spirit of the classical Calculus of Variations,
regularity results should be chased by looking at special classes of functionals, and
thinking of relevant model examples, therefore limiting the degree of generality
one wants to achieve. One of such relevant classes, for which a general and rather
complete theory is now available, is with no doubt the one of functionals with
so called p(x)-growth, i.e. functionals of the type (2.1), satisfying the following,
“variable exponent version” of the growth conditions in (2.11):
(7.1) ν|z|
p(x)
≤ F (x, y, z) ≤ L(1 + |z|
p(x)
) .
The exponent function p : Ω → (1, ∞) will be here considered to be continuous and
satisfying
(7.2) 1 < γ
1
≤ p(x) ≤ γ
2
< ∞ .
The clear prototype is the functional D
p(x)
, we already met in Section 6:
D
p(x)
(v) :=
Z
Ω
|Dv|
p(x)
dx .
The assumption (7.2) clearly serves to ensure that D
p(x)
keeps far from either the
total variation functional [287], and the so called ∞-Laplacean [20]. This energy

36 GIUSEPPE MINGIONE
shows up when considering a number of models from Mathematical Physics: Ho-
mogenization of strongly anisotropic material, as pioneered by Zhikov [307, 311],
Electro-rheological fluids as modelled by Rajagopal & R˚uˇziˇcka [270, 274], tempera-
ture dependent viscosity fluids, as again conceived by Zhikov [310], Image processing
models by Chen & Levine & Rao as in [58]. More generally, a functional as D
p(x)
serves when modelling physical situation with strong anisotropicity, the anisotropic
nature of the situation described by the appearance of the x-variable in the growth
exponent. Here I will confine myself to report the basic regularity results for mini-
mizers of the functional D
p(x)
available in the literature up to now. The same will
apply to solutions to the related Euler-Lagrange system
(7.3) div(p(x)|Du|
p(x)−2
Du) = 0 .
The results I am going to present are also valid for more general functionals, equa-
tions, and systems, with “p(x)-growth”, provided suitable distinctions between the
scalar and vectorial case are done; for this I refer to [5, 6, 8, 66]. I emphasize here
that, when referring to D
p(x)
and the system (7.3), I shall consider the problem in
the general vectorial case u : Ω → R
N
, and N ≥ 1. We have seen in the previous
section that, when considering non-autonomous functionals with (p, q)-growth con-
ditions, the regularity of minimizers depends on a subtle interaction between the
gap q/p and the regularity of the integrand F (x, z) with respect to the x-variable.
In particular, condition (6.15) tells us that, in a certain sense, the H¨older continuity
with respect to x serves to “re-balance” the distance between p and q created by
the very fact that x varies, compare Paragraph 6.5. In the case of the functional
D
p(x)
the key idea is to think that on small domains, say small balls B
s
⊂ Ω, the
gap q/p of the integrand |z |
p(x)
can be made arbitrarily near 1, since
q := sup
B
s
p(x) , p := inf
B
s
p(x) ,
and the function p(x) is continuous. Therefore, since proving local regularity results
is something that can be done reducing on arbitrarily small open subset of Ω,
and then concluding with a standard covering argument, wee see at once that
for the functional D
p(x)
, if thinking of condition (6.15), any, possibly small, H¨older
exponent α > 0 suffices in order to get regularity of minima in the sense of Theorem
6.6. We have actually much more: due to the peculiar structure of D
p(x)
, condition
(6.15) admits a borderline case, that is the so called log-continuity assumption, first
introduced by Zhikov [309] to treat the Lavrentiev Phenomenon related to D
p(x)
.
This goes as follows: let me call ω(·) the modulus of continuity of the exponent
function p(x), that is
ω(s) := sup
B
s
⊂⊂Ω
x,y∈B
s
|p(x) − p(y)| .
Then the log-continuity assumption prescribes that
(7.4) lim sup
s→0
ω(s) log
1
s
= L < ∞ .
Such an assumption turns out to be crucial: Zhikov proved that the failure of (7.4)
is a possible cause of discontinuities of minima [309], see also [180]. On the other
hand, assuming (7.4) allows to prove higher integrability of minimizers, that is,
with u ∈ W
1,1
(Ω, R
N
) being a local minimizer of D
p(x)
, there exists a δ > 0 such
that
(7.5)
Z
Ω
|Du|
p(x)(1+δ)
dx < ∞ .
Moreover, Zhikov proved that (7.4) is a sort of “universal condition”, linking reg-
ularity of solutions and minima, to the structure of the spaces L
p(x)
(Ω, R
N
), and

REGULARITY IN THE DARK SIDE 37
to Lavrentiev Phenomenon for the functional D
p(x)
. Let’s give a rapid look at the
situation. The space L
p(x)
(Ω, R
N
) is defined as
(7.6)
L
p(x)
(Ω, R
N
) :=
½
v : Ω → R
N
: v is measurable and
Z
Ω
|v|
p(x)
dx < ∞
¾
,
and once equipped with the Luxemburg norm
kvk
L
p(x)
(Ω,R
N
)
:= inf
½
λ > 0 :
Z
Ω
¯
¯
¯
v
λ
¯
¯
¯
p(x)
dx ≤ 1
¾
it becomes a Banach space. The space L
p(x)
(Ω, R
N
) is an instance, actually the
most important and popular one, of the so called Orlicz-Musielak spaces [263, 85].
Accordingly, the generalized W
1,p(x)
(Ω, R
N
) space is defined by
W
1,p(x)
(Ω, R
N
) :=
n
v ∈ L
p(x)
(Ω, R
N
) : Dv ∈ L
p(x)
(Ω, R
nN
)
o
,
where Dv obviously denotes the distributional gradient of the map v. This also
becomes a Banach space with norm defined by
kvk
W
1,p(x)
(Ω,R
N
)
:= kvk
L
p(x)
(Ω,R
N
)
+ kDvk
L
p(x)
(Ω,R
nN
)
.
Zhikov essentially proved that (7.4) implies the absence of Lavrentiev Phenomenon
for the functional D
p(x)
, according to the approximation property in (6.16), and he
also proved that the convolution/mollification operator is bounded when assuming
condition (7.4). Subsequently, a massive quantity of interesting contributions have
been given on the spaces L
p(x)
(Ω, R
N
); there is no room here to give account
of this, and I will refer to the recent, excellent surveys [86, 276]. I hereby just
want to mention the results obtained by Diening & R˚uˇziˇcka [87], who proved that
singular integral operators are bounded in L
p(x)
if and only if (7.4) is satisfied,
while boundedness results in L
p(x)
for fractional maximal-type operators have been
obtained by Kokilashvili & Samko [196]; see also [85] for more on Harmonic Analysis
in L
p(x)
-spaces.
In a series of papers, Acerbi, Coscia, and myself, investigated the regularity
properties of local minimizers of D
p(x)
, when assuming condition (7.4) and/or suit-
able reinforcements [5, 6, 7, 8, 66]. Further contributions to regularity, also for
non-variational situations, and in the parabolic case, are those in [11, 118, 119, 59,
10, 17, 15, 16]. I am starting with the H¨older continuity of minimizers.
Theorem 7.1. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional D
p(x)
,
under the assumption
(7.7) lim sup
s→0
ω(s) log
1
s
= 0 .
Then u ∈ C
0,α
loc
(Ω, R
N
), for every α < 1.
The proof of the previous result in the scalar case is contained in [5]; the one
in the vectorial case goes along the lines of the scalar one, keeping into account of
the estimates in [66]. It is to be noted that assuming only (7.4) in the scalar case
N = 1, then one can prove that u ∈ C
0,β
loc
(Ω), for some (small) β > 0, see [11]. This
is not by chance; indeed by [5] we still infer that: For every ˜α ∈ (0, 1) there exists
ε ≡ ε(˜α) > 0 such that if
lim sup
s→0
ω(s) log
1
s
≤ ε ,
then u ∈ C
0,˜α
loc
(Ω). In other words, controlling the oscillations of the exponent
function p(x) allows to control the degree of regularity of local minimizers.

38 GIUSEPPE MINGIONE
In order to get the H¨older continuity of the gradient it is unavoidable to assume
that p(x) is itself H¨older continuous, that is
(7.8) ω(s) ≤ Ls
α
, α > 0 ,
which is obviously stronger that (7.7). The result, taken from [66], is the following:
Theorem 7.2. Let u ∈ W
1,p
(Ω, R
N
) be a local minimizer of the functional D
p(x)
,
under the assumption (7.8). Then Du ∈ C
0,β
loc
(Ω, R
nN
), for some β < α.
The previous result is sharp in the sense that if p(x) is not H¨older continuous
then the gradient is not even continuous in general, as shown in [179]. Moreover, it
is not possible to get that Du ∈ C
0,β
loc
(Ω, R
nN
), for every β < 1, the counterexample
already working in the case where p(x) is a constant function [302].
The proof of Theorems 7.1-7.2 is based on a delicate combination of ingredients:
a careful localization argument starting from the higher integrability property (7.5);
a perturbation-in-the-exponent method, build on a combined use of Reverse H¨older
inequalities and estimates in the space L log L. The regularity of local minima of
D
p(x)
is indeed obtained by comparison with minimizers of (2.16), for a suitable
choice of the fixed exp onent p. The variations in the exponent naturally make
quantities as
Z
|Du|
p(x)(1+δ )
log(e + |Du|) dx ,
appear, and these have to estimated very carefully in order to get the result under
the optimal assumption (7.7). At this stage another crucial role is played by the
so-called “stability of the estimates” for solutions to the p-Laplacean system (2.17):
all the constants involved in the local C
0,1
and C
1,α
estimates of solutions to (2.17),
including α, do not blow-up, neither degenerate, as long as p varies in a compact
subset of (1, ∞). For this I again recommend Manfredi’s thesis [232], or [122].
The local regularity results for minimizers of Theorems 7.1-7.2 immediately extend
to solutions to the system (7.3), as I myself showed in the lectures of the Paseky
course, the proof being actually simpler. The proof of the C
0,α
regularity can be
obtained as a corollary of the results in [8], while the proof of the C
1,β
result for
general equations is unfortunately not explicitly written anywhere, but it easily
follows from the arguments in [66], where on the other hand the model system
(7.3) is obviously covered. Again, Theorems 7.1 and 7.2 extend to more general
functionals with p(x)-growth i.e. functionals of the type (2.1) whose integrand is in
a suitable sense controlled by |Du|
p(x)
; see also [5, 107].
In [8], Acerbi and myself have given yet another proof of Theorem 7.1, that
eventually follows as a particular case of the next Calder´on-Zygmund type result.
Let me consider the following non-homogeneous p(x)-Laplacean system:
(7.9) div(p(x)|Du|
p(x)−2
Du) = div(|F |
p(x)−2
F ) ,
where F : Ω → R
nN
is a prescribed vector field such that
Z
Ω
|F |
p(x)
dx < ∞ .
By a weak solution to (7.9) I mean, of course, a map u ∈ W
1,p(x)
(Ω, R
N
) such that
Z
Ω
p(x)|Du|
p(x)−2
DuDϕ dx =
Z
Ω
|F (x)|
p(x)−2
F (x)Dϕ dx
for every test function ϕ ∈ W
1,p(x)
(Ω, R
N
) with compact support in Ω. For the
related existence theory the reader should look at the work of R˚uˇziˇcka [274]. We
have

REGULARITY IN THE DARK SIDE 39
Theorem 7.3. Let u ∈ W
1,p(x)
(Ω, R
N
) be a weak solution to the non-homogeneous
p(x)-Laplacean system (7.9), under the assumption (7.7). Then
(7.10) |F |
p(x)
∈ L
q
loc
(Ω) =⇒ |Du|
p(x)
∈ L
q
loc
(Ω) , ∀ q > 1 .
The previous result is again sharp [309], in that without assuming at least (7.4)
the statement is not true; for more precise comments see [8], Remark 2. Theorem
7.1 follows with the choice F ≡ 0, and then applying Sobolev embedding theorem.
We are also able to provide a local a priori estimates for the gradient Du in terms
of certain natural Reverse H¨older inequalities, see Theorem 2 in [8]. In the case
p(x) ≡ constant, Theorem 7.3 is due to T. Iwaniec [189] in the scalar case N = 1,
and to DiBenedetto & Manfredi [83] for the case N > 1; for L
q
-estimates for the
p-Laplacean operator see also the paper by Caffarelli & Peral [51]. The proof of
Theorem 7.3 yields anyway new results already in this classical case, in that we are
able to treat also a class of non linear, degenerate elliptic equations with p-growth
in divergence form. The methods in [8] readily extend to cover more general right
hand sides for the p(x)-Laplacean system, as, for instance
div(p(x)|Du|
p(x)−2
Du) = div F .
Let me conclude going back to the case of a fixed, non-variable growth exponent
p. Very recently, in [9], the elliptic results in [189] and [83] have been extended
to a large class of parabolic operators whose model type is the non-homogeneous,
parabolic p-Laplacean operator
(7.11) u
t
− div(|Du|
p−2
Du) = div(|F |
p−2
F ) ,
under the assumption
(7.12) p >
2n
n + 2
.
In this situation the system (7.11) is considered in a cylindrical domain C :=
Ω × [0, T ), where T > 0 and Ω ⊂ R
n
is, as usual, a bounded domain, while the
solution u is with no loss of generality considered in the space
u ∈ C
0
((0, T ); L
2
(Ω, R
N
)) ∩ L
p
(0, T ; W
1,p
(Ω, R
N
)) ,
and N ≥ 1; finally F ∈ L
p
(C, R
nN
) . Under the previous assumptions Acerbi &
myself proved the following analog of (7.10):
(7.13) |F |
p
∈ L
q
loc
(C) =⇒ |Du|
p
∈ L
q
loc
(C) ∀ q > 1 .
Moreover, we were also able to provide precise local estimates of Reverse-H¨older
type, bounding the L
pq
norm of Du in terms of that of F , and the L
p
norm of Du
itself. Such estimates are natural in that, in the homogeneous case F ≡ 0, by letting
q % ∞, we recover from them the classical local C
0,1
-estimates of DiBenedetto and
Friedman [82]. The lower bound in (7.12) is necessary in order to obtain the result
in (7.13). The new technical contribution of [9] consists of providing a method
which is completely free of Harmonic Analysis tools. Indeed in the papers [189, 83]
crucial use is made of various maximal operators; this is not possible in the case
of the systems as (7.11). Indeed all estimates must be carried out according to
the “intrinsic geometry viewpoint” of DiBenedetto [81], and therefore on parabolic
cylinders whose size depends on the solution itself. Such cubes are a priori arbitrary,
and therefore not related to any fixed maximal operator. On the contrary, we rely on
a new method involving several diffferent ingredients. For instance, we are directly
arguing on certain Calder´on-Zygmund type coverings of the level sets of the gradient
Du, which are locally adapted to the solution, and use them in combination with
the C
0,1
estimates available in the case of the homogeneous parabolic p-Laplacean
system [82], that is (7.11) with F ≡ 0. Moreover, since we are not using any

40 GIUSEPPE MINGIONE
maximal type operator, we cannot use the so called “good-λ-inequality” principle
as in [8]; on the contrary, we introduce an analog version of that, working again on
Calder´on-Zygmund cubes directly: we called it the “large-M-inequality” principle.
This time the method is flexible enough to include more general systems with
possibly discontinuous coefficients of the type
u
t
− div(a(x, t)|Du|
p−2
Du) = div(|F |
p−2
F ) ,
where ν ≤ a(x, t) ≤ L may be discontinuous in a suitable VMO/BMO fashion; this
extends previous, elliptic results of Kinnunen & Zhou [199], where again maximal
operators are crucially employed. Studying Calder´on-Zygmund type estimates for
equations with discontinuous coefficients has been the object of intensive investi-
gation at length: see [60, 88, 42], and references. Moreover the method extends to
all degenerate/singular parabolic equations in divergence form of the type
u
t
− div a(x, t, Du) = div(|F |
p−2
F ) ,
where the vector field a satisfies the assumptions in (2.18), suitably recast for the
case under consideration, but just requiring continuity dependence with respect to
(x, t), and not H¨older continuity.
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Dipartimento di Matematica, Universit
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