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An extremal property of the inf- and
sup-convolutions regarding the Strong Maximum
Principle∗
Vladimir V. Goncharov†and Telma J. Santos
CIMA-UE, rua Rom˜ao Ramalho 59, 7000-671, ´
Evora, Portugal‡
Abstract
In this paper we continue investigations started in [6] concerning the
extension of the variational Strong Maximum Principle for lagrangeans
depending on the gradient through a Minkowski gauge. We essentially
enlarge the class of comparison functions, which substitute the identical
zero when the lagrangean is not longer strictly convex at the origin.
1 Introduction
The Strong Maximum Principle, a well known property of the elliptic partial
differential equations (see, e.g., [5, 8] and the bibliography therein), can be for-
mulated in the variational setting as was done by A. Cellina in 2002. Extending
the main result in [3] we consider the integral functional
ZΩ
f(ρF(∇u(x))) dx, (1.1)
where Ω ⊂Rnis an open bounded connected domain; f:R+→R+∪ {+∞},
f(0) = 0, is a lower semicontinuous convex function; F⊂Rnis a convex
closed bounded set with 0 ∈intF(interior of F), and ρF(·) is the Minkowski
functional (gauge function) associated to F,
ρF(ξ) := inf {λ > 0 : ξ∈λF }. (1.2)
∗Work is financially supported by FCT (Funda¸c˜ao para Ciˆencia e Tecnologia, Portugal),
the project ”An´alise Variacional: Teoria e Aplica¸c˜oes”, PTDC/MAT/111809/2009; and Fi-
nanciamento Program´atico Especial do CIMA-UE (Centro de Investiga¸c˜ao em Matem´atica e
Aplica¸c˜oes da Universidade de ´
Evora)
†Corresponding author. Tel.: +351 936514934; Fax: +351 266745393
‡E-mail: goncha@uevora.pt (V. Goncharov); tjfs@uevora.pt (T. Santos)
In the traditional sense, the Strong Maximum Principle (SMP) for (1.1)
means that there is no a nonconstant continuous minimizer of this functional
on u0(·) + W1,1
0(Ω) (with some Sobolev function u0(·)), admitting its mini-
mal (maximal) value in Ω. In the case of a rotationally invariant lagrangean
(ρF(ξ) = kξk) and n > 1 it was proven in [3] that this property is valid if and
only if the function f(·) is strictly convex and smooth at the origin, or, in other
words, if the equalities
∂f ∗(0) = {0}(1.3)
and
∂f (0) = {0}(1.4)
hold. Here, as usual, ∂ f stands for the subdifferential of the function f(·)
in the sense of Convex Analysis, and f∗(·) is the Legendre-Fenchel tranform
(conjugate) of f(·). Observe that in the case n= 1 the smoothness of f(·)
at zero (assumption (1.4)) is not necessary, and the SMP is valid unless the
function f(·) is not affine near the origin (the latter condition already contains
(1.3)).
In [6] we proved that under the same hypotheses on f(·) the Strong Max-
imum Principle remains valid for a general functional (1.1), where the gauge
Fis not assumed to be either rotund, smooth or symmetric. Furthermore, we
tried to extend the SMP to the case when the condition (1.3) fails.
Since the SMP can be equivalently reformulated as a comparison property:
if a continuous nonnegative (nonpositive)
minimizer u(·)of the functional (1.1) on
u0(·) + W1,1
0(Ω) touches zero at some point
x∗∈Ωthen necessarily u(·)≡0on Ω,(1.5)
it is obviously violated whenever the lagrangean is no longer strictly convex at
the origin (see also [2]). Nevertheless, we emphasized a class Cof continuous
functions, which being themselves solutions of the variational problem can sub-
stitute in some sense the identical zero in the property (1.5). These functions
(further called test, or comparison, functions) depend certainly on the subdif-
ferential ∂f ∗(0) and reduce to constants when ∂f ∗(0) reduces to the singleton
{0}. In the place of the null-function in (1.5), clearly, any constant can stand.
In the case ∂f ∗(0) 6={0}so extended Strong Maximum Principle for a test
function ˆu(·)∈Ccan be given as follows:
each continuous minimizer of (1.1) on
u0(·) + W1,1
0(Ω) such that u(x)≥ˆu(x) (respectively,
u(x)≤ˆu(x) ), x∈Ω, having the same points
of local minimum (respectively, local maximum)
as ˆu(·)should coincide with ˆu(·)everywhere on Ω. (1.6)
2
Observe that all the functions ˆu(·)∈Care written in terms of the polar set
F0. Namely, in the simplest case of unique local minimum (local maximum)
point x0∈Ω the functions ˆu+
x0,µ (x) := µ+aρF0(x−x0) and ˆu−
x0,µ (x) :=
µ−aρF0(x0−x) belong to Cfor each real µ. Here a:= sup ∂f ∗(0). If instead
∂f ∗(0) = {0}(equivalently, a= 0) then we can take x0=x∗where x∗is an
arbitrary ”floating” point from Ω (see (1.5)), and we arrive at the traditional
SMP although the property (1.6) is not formally applicable.
In [6] also a ”multipoint” version of the Strong Maximum Principle was
established when the comparison function ˆu(·) is the lower (upper) envelope of
a finite number of functions ˆu+
x0,µ (·) (respectively, ˆu−
x0,µ (·)) for various x0∈Ω
and µ∈R. Notice that (1.6) takes place only for convex domains Ω ⊂Rn(or,
at least, under a kind of star-shapeness hypothesis that can not be removed, see
[6]).
Another restriction, under which validity of the property (1.6) was proven, is
smoothness of the gauge function ρF(·), or, equivalently, rotundity of the polar
set F0. In fact, one of the tools we use in the proofs is the so named modulus
of rotundity
MF0(r;α, β) := inf {1−ρF0(ξ+λ(η−ξ)) :
ξ, η ∈∂F 0,ρF0(ξ−η)≥r,α≤λ≤β, (1.7)
which is strictly positive for all r > 0 and all 0 < α ≤β < 1 whenever F0is
rotund.
In this section we essentially enlarge the class Cenvolving the infinite (con-
tinuous) envelopes of the functions ˆu±
x0,µ (·) by such a way that the generalized
SMP gets an unique extremal extension principle and unifies both properties
(1.5) and (1.6). Namely, given an arbitrary function θ(·) defined on a closed
subset Γ ⊂Ω and satisfying a natural slope condition w.r.t. Fwe prove in
Section 3 that the inf-convolution
u+
Γ,θ (x) := inf
y∈Γ{θ(y) + aρF0(x−y)}(1.8)
(respectively, the sup-convolution
u−
Γ,θ (x) := sup
y∈Γ
{θ(y)−aρF0(y−x)}) (1.9)
is the only continuous minimizer u(·) of the functional (1.1) on u0(·)+ W1,1
0(Ω)
such that u(x) = θ(x) on Γ and u(x)≥u+
Γ,θ (x) (respectively, u(x)≤u−
Γ,θ (x)),
x∈Ω. The domain Ω is always assumed to be convex.
2 Preliminaries. Auxiliary statements
In what follows we assume that
a:= sup ∂f ∗(0) >0,
3
and so the second Cellina’s hypothesis (1.4) is automatically fulfilled. Further-
more, we introduce the nondecreasing upper semicontinuous function
ϕ(t) := sup ∂f ∗(t) .
So ϕ(0) = aand ϕ(t)<+∞on the interior of the domain
domf∗:= t∈R+:f∗(t)<+∞.
The version of SMP we wish to prove is essentially based on the following local
estimates of continuous minimizers of (1.1) obtained in [6] by using the dual
properties of convex sets (see, e.g., [9] or [7]), being themselves an interesting
result of Convex Analysis.
Theorem 1 Given an open bounded region Ω⊂Rn,n≥1, and a continuous
admissible minimizer ¯u(·)of the functional (1.1) on u0(·) + W1,1
0(Ω), assume
a point ¯x∈Ωand real numbers β > 0and µto be such that
¯u(x)≥µ∀x∈¯x−β F 0⊂Ω
and
¯u(¯x)> µ +aβ.
Then for some η > 0the inequality
¯u(x)≥µ+ϕ(η) (β−ρF0(¯x−x)) (2.1)
holds for all x∈¯x−βF 0.
Simmetrically, if a point ¯x∈Ωand numbers β > 0and µare such that
¯u(x)≤µ∀x∈¯x+β F 0⊂Ω
and
¯u(¯x)< µ −aβ,
then there exists η > 0such that
¯u(x)≤µ−ϕ(η) (β−ρF0(x−¯x)) (2.2)
for all x∈¯x+β F 0.
Roughly speaking, the statement above means that for each continuous ad-
missible minimizer ¯u(·) of (1.1) and for each point ¯x∈Ω, which is not local
extremum for ¯u(·), the deviation of ¯u(·) from the extremal level can be con-
trolled near ¯xby an affine transformation of the dual Minkowski gauge (see (2.1)
and (2.2)). Recall that admissible minimizers are those giving finite values to
functional (1.1).
In the case a > 0 (it is our standing assumption along with the paper) we
have the following simple consequence of this theorem.
4
Corollary 1 Given Ω⊂Rn,n≥1, and ¯u(·)as in Theorem 1 let us assume
that for some x0∈Ωand δ > 0
¯u(x)≥¯u(x0) + aρF0(x−x0)∀x∈x0+δ F 0⊂Ω.(2.3)
Then
¯u(x) = ¯u(x0) + aρF0(x−x0)∀x∈x0+δ
kFk kF0k+ 1F0.
Similarly, if in the place of (2.3)
¯u(x)≤¯u(x0)−aρF0(x0−x)∀x∈x0−δ F 0⊂Ω (2.4)
then
¯u(x) = ¯u(x0)−aρF0(x0−x)∀x∈x0−δ
kFk kF0k+ 1F0.
Here kFk:= sup {kξk:ξ∈F}.
As standing hypotheses in what follows we assume that F⊂Rnis a convex
closed bounded set, 0 ∈intF, with smooth boundary (the latter means that the
Minkowski functional ρF(ξ) is Fr´echet differentiable at each ξ6= 0), and that
Ω⊂Rnis an open convex bounded region.
Let us consider an arbitrary nonempty closed subset Γ ⊂Ω and a function
θ: Γ →Rsatisfying the slope condition:
θ(x)−θ(y)≤aσF(x−y)∀x, y ∈Γ, (2.5)
where
σF(ξ) := sup
v∈F
hv, ξi
is the support function of F(h·,·i means the inner product in Rn). It is well
known that
•F00=F;
•σF(ξ) = ρF0(ξ) whenever ξ∈F0;
•the polar set F0is rotund (see Section 1).
We will use also the following property of the gauge function:
1
kFkkξk ≤ ρF(ξ)≤
F0
kξk.(2.6)
Let us define now inf- and sup-convolutions of θ(·) with the gauge function
aρF0(·) by the formulas (1.8) and (1.9). Observe first that the function u±
Γ,θ (·)
5
is the minimizer of (1.1) on u±
Γ,θ (·) + W1,1
0(Ω). Indeed, it is obviously Lipschitz
continuous on Ω, and for its (classic) gradient ∇u±
Γ,θ existing by Rademacher’s
theorem we have that
∇u±
Γ,θ (x)∈∂cu±
Γ,θ (x)⊂aF
for a.e. x∈Ω (see [4, Theorem 2.8.6]). Here ∂cstands for the Clarke’s subdif-
ferential of a (locally) Lipschitzean function. Consequently,
fρF∇u±
Γ,θ (x)= 0
a.e. on Ω, and the function u±
Γ,θ (·) gives to (1.1) the minimal possible value
zero.
Due to the slope condition (2.5) it follows that u±
Γ,θ (x) = θ(x) for all x∈
Γ. Moreover, u±
Γ,θ (·) is the (unique) viscosity solution of the Hamilton-Jacobi
equation
±(ρF(∇u(x)) −a) = 0, u|Γ=θ,
(see, e.g., [1]).
Notice that Γ can be a finite set, say {x1, x2, ..., xm}.In this case θ(·)
associates to each xia real number θi,i= 1, ..., m, and the condition (2.5)
slightly strengthened (by assuming that the inequality in (2.5) is strict for
xi6=xj) means that all the simplest test functions θi+aρF0(x−xi) (respec-
tively, θi−aρF0(xi−x)) are essential (not superfluous) in constructing the
respective lower or upper envelope. Then the extremal property established
below is reduced to the extended SMP (1.6) (see [6, Theorem 6]).
On the other hand, if θ(·) is a Lipschitz continuous function defined on a
closed convex set Γ ⊂Ω with nonempty interior then (2.5) holds iff
∇θ(x)∈aF
for almost each (a.e.) x∈Γ. This immediately follows from Lebourg’s mean
value theorem (see [4, p. 41]) recalling the properties of the Clarke’s subdiffer-
ential and from the separability theorem.
Certainly, the mixed (discrete and continuous) case can be considered as
well, and all the situations are unified by the hypothesis (2.5).
In the particular case θ≡0 ((2.5) is trivially fulfilled) the function u+
Γ,θ (x)
is nothing else than the minimal time necessary to achieve the closed set Γ from
the point x∈Ω by trajectories of the differential inclusion with the constant
convex right-hand side
−a˙x(t)∈F0, (2.7)
while −u−
Γ,θ (x) is, contrarily, the minimal time, for which trajectories of (2.7)
arrive at xstarting from a point of Γ. Furthermore, if F=Bis the closed unit
ball centered at the origin then the gauge function ρF0(·) is the euclidean norm
in Rn, and we have
u±
Γ,θ (x) = ±adΓ(x)
6
where dΓ(·) means the distance from a point to the set Γ.
Proving the main theorem below we essentially use the following simple prop-
erty of extremums in (1.8) and (1.9), which generalizes a well-known property
of metric projections (see [6, Lemma 1]).
Proposition 1 Given an arbitrary nonempty closed set Γ⊂Rnand a real-
valued function θ(·)defined on Γlet us assume that for x∈Rn\Γthe minimum
of y7→ θ(y)+aρF0(x−y)(respectively, the maximum of y7→ θ(y)−aρF0(y−x))
on Γis attained at some point ¯y∈Γ. Then ¯yis also a minimizer of y7→
θ(y) + aρF0(xλ−y)(respectively, the maximizer of y7→ θ(y)−aρF0(y−xλ))
for all λ∈[0,1],where xλ:= λx + (1 −λ) ¯y.
3 Generalized Strong Maximum Principle
Now we are ready to deduce the extremal property of the functions u±
Γ,θ (·)
announced above.
Theorem 2 Under all the standing hypotheses formulated in the previous sec-
tion let us assume that a continuous admissible minimizer ¯u(·)of functional
(1.1) on u0(·) + W1,1
0(Ω) is such that
(i) ¯u(x) = u+
Γ,θ (x) = θ(x)∀x∈Γ;
(ii) ¯u(x)≥u+
Γ,θ (x)∀x∈Ω.
Then ¯u(x)≡u+
Γ,θ (x)on Ω.
Simmetrically, if a continuous admissible minimizer ¯u(·)satisfies the con-
ditions
(i)0¯u(x) = u−
Γ,θ (x) = θ(x)∀x∈Γ;
(ii)0¯u(x)≤u−
Γ,θ (x)∀x∈Ω,
then ¯u(x)≡u−
Γ,θ (x)on Ω.
Proof. Let us prove the first part of Theorem only since the respective change-
ments in the symmetric case are obvious.
Given a continuous admissible minimizer ¯u(·) satisfying conditions (i) and
(ii) we suppose, on the contrary, that there exists ¯x∈Ω\Γ with ¯u(¯x)> u+
Γ,θ (¯x).
Let us denote by
Γ+:= nx∈Ω : ¯u(x) = u+
Γ,θ (x)o
7
and claim that
u+
Γ,θ (x) = inf
y∈Γ+{¯u(y) + aρF0(x−y)}(3.1)
for each x∈Ω. Indeed, the inequality ”≥” in (3.1) is obvious because Γ+⊃Γ
and ¯u(y) = θ(y), y∈Γ. On the other hand, given x∈Ω take an arbitrary
y∈Γ+and due to the compactness of Γ we find y∗∈Γ such that
¯u(y) = θ(y∗) + aρF0(y−y∗) . (3.2)
Then, by triangle inequality,
¯u(y) + aρF0(x−y)≥θ(y∗) + aρF0(x−y∗)≥
≥u+
Γ,θ (x) . (3.3)
Passing to infimum in (3.3) we prove the inequality ”≤” in (3.1) as well.
Since for arbitrary x, y ∈Γ+and for y∗∈Γ satisfying (3.2) we have
¯u(x)−¯u(y) = u+
Γ,θ (x)−u+
Γ,θ (y)≤aρF0(x−y∗)−aρF0(y−y∗)≤
≤aσF(x−y) , (3.4)
we can extend the function θ: Γ →Rto the (closed) set Γ+⊂Ω by setting
θ(x) = ¯u(x) , x∈Γ+,
and all the conditions on θ(·) remain valid (see (3.4) and (3.1)). So, without
loss of generality we can assume that the strict inequality
¯u(x)> u+
Γ,θ (x) (3.5)
holds for all x∈Ω\Γ6=∅.
Notice that the convex hull K:= co Γ is the compact set contained in Ω
(due to the convexity of Ω). Let us choose now ε > 0 such that K±εF 0⊂Ω
and denote by
δ:= 2εMF02ε
∆;ε
ε+ ∆,∆
ε+ ∆>0,
where MF0is the modulus of rotundity associated to F0(see (1.7)) and
∆ := sup
ξ,η∈Ω
ρF0(ξ−η)
is the ρF0-diameter of the region Ω. Similarly as in [6] (see Step 1 in the proof
of Theorem 5) we show that
ρF0(y1−x) + ρF0(x−y2)−ρF0(y1−y2)≥δ(3.6)
whenever y1, y2∈Γ and x∈ΩK+εF 0∪K−εF 0. Indeed, we obvi-
ously have
ε≤ρ1:= ρF0(y1−x)≤∆ and ε≤ρ2:= ρF0(x−y2)≤∆,
8
and, consequently,
λ:= ρ2
ρ1+ρ2
∈ε
ε+ ∆,∆
ε+ ∆. (3.7)
Setting ξ1:= y1−x
ρ1and ξ2:= x−y2
ρ2we can write
ξ1−ξ2=1
ρ1
+1
ρ2 ρ2
ρ1+ρ2
y1+ρ1
ρ1+ρ2
y2−x,
and hence
ρF0(ξ1−ξ2)≥1
ρ1
+1
ρ2ε≥2ε
∆. (3.8)
On the other hand,
ρF0(y1−x) + ρF0(x−y2)−ρF0(y1−y2)
= (ρ1+ρ2)1−ρF0ρ1
ρ1+ρ2
ξ1+ρ2
ρ1+ρ2
ξ2≥
≥2ε[1 −ρF0(ξ1+λ(ξ2−ξ1))] . (3.9)
Combining (3.9), (3.8), (3.7) and the definition of the rotundity modulus (1.7)
we arrive at (3.6).
Let us fix ¯x∈ΩΓ and ¯y∈Γ such that
u+
Γ,θ (¯x) = θ(¯y) + aρF0(¯x−¯y) .
Then by Proposition 1 the point ¯yis also a minimizer on Γ of the function
y7→ θ(y) + aρF0(xλ−y),where xλ:= λ¯x+ (1 −λ) ¯y,λ∈[0,1], i.e.,
u+
Γ,θ (xλ) = θ(¯y) + aρF0(xλ−¯y) . (3.10)
Define now the Lipschitz continuous function
¯v(x) := max {¯u(x),min {θ(¯y) + aρF0(x−¯y),
θ(¯y) + a(δ−ρF0(¯y−x))}} (3.11)
and claim that ¯v(·) minimizes the functional (1.1) on the set ¯u(·) + W1,1
0(Ω).
In order to prove this we observe first that for each x∈Ω, x /∈K±εF0, and
for each y∈Γ,by the slope condition (2.5) and by (3.6), the inequality
θ(y) + aρF0(x−y)−θ(¯y) + aρF0(¯y−x)
≥a(ρF0(¯y−x) + ρF0(x−y)−ρF0(¯y−y)) ≥aδ (3.12)
holds. Passing to infimum in (3.12) for y∈Γ and taking into account the basic
assumption (ii),we have
¯u(x)≥inf
y∈Γ{θ(y) + aρF0(x−y)}
≥θ(¯y) + a(δ−ρF0(¯y−x)) ,
9
and, consequently, ¯v(x) = ¯u(x) for all x∈Ω\K+εF 0∪K−εF 0. In
particular, ¯v(·)∈¯u(·) + W1,1
0(Ω). Furthermore, setting
Ω0:= {x∈Ω : ¯v(x)6= ¯u(x)},
by the well known property of the support function, we have ∇¯v(x)∈aF for
a.e. x∈Ω0, while ∇¯v(x) = ∇¯u(x) for a.e. x∈ΩΩ0. Then
ZΩ
f(ρF(∇¯v(x))) dx =Z
ΩΩ0
f(ρF(∇¯u(x))) dx
≤ZΩ
f(ρF(∇¯u(x))) dx ≤ZΩ
f(ρF(∇u(x))) dx
for each u(·)∈¯u(·) + W1,1
0(Ω).
Finally, setting
µ:= min (ε, δ
(kFk kF0k+ 1)2),
we see that the minimizer ¯v(·) satisfies the inequality
¯v(x)≥θ(¯y) + aρF0(x−¯y) (3.13)
on ¯y+µkFk
F0
+ 1F0.Indeed, it follows from (2.6) that
ρF0(x−¯y) + ρF0(¯y−x)≤µkFk
F0
+ 12≤δ
whenever ρF0(x−¯y)≤µkFk
F0
+ 1, implying that the minimum in (3.11)
is equal to θ(¯y) + aρF0(x−¯y). Since, obviously, ¯v(¯y) = θ(¯y), applying Corol-
lary 1 we deduce from (3.13) that
¯v(x) = θ(¯y) + aρF0(x−¯y)
for all x∈¯y+µF 0⊂K+εF 0⊂Ω. Taking into account (3.11), we have
¯u(x)≤θ(¯y) + aρF0(x−¯y) , x∈¯y+µF 0. (3.14)
On the other hand, for some λ0∈(0,1] the points xλ, 0 ≤λ≤λ0, belong to
¯y+µF 0. Combining now (3.14) for x=xλwith (3.10) we obtain
¯u(xλ)≤u+
Γ,θ (xλ)
and hence (see the hypothesis (ii))
¯u(xλ) = u+
Γ,θ (xλ) ,
0≤λ≤λ0, contradicting (3.5).
10
Acknowledgements.The authors thank the Portuguese Foundation for Sci-
ence and Technologies (FCT), the Portuguese Operational Programme for Com-
petitiveness Factors (COMPETE), the Portuguese Strategic Reference Frame-
work (QREN) and the European Regional Development Fund (FEDER) for the
financial support.
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