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arXiv:1711.08339v1 [math.AP] 22 Nov 2017

FREE BOUNDARY PROBLEMS INVOLVING SINGULAR

WEIGHTS

JIMMY LAMBOLEY, YANNICK SIRE, AND EDUARDO V. TEIXEIRA

Abstract. In this paper we initiate the investigation of free bound-

ary minimization problems ruled by general singular operators with A2

weights. We show existence and boundedness of minimizers. The key

novelty is a sharp C1+γregularity result for solutions at their singular

free boundary points. We also show a corresponding non-degeneracy

estimate.

Contents

1. Introduction 1

2. Mathematical set-up 3

3. Existence and local boundedness 6

4. Compactness 7

Homogenization 8

5. C1+γregularity at the free boundary 11

6. Nondegeneracy and weak geometry 13

References 15

Aknowledgement: The ﬁrst two authors would like to thank the hospi-

tality of the Univ. Federal do Ceara in Fortaleza, where this work initiated,

and the Brazilian-French Network in Mathematics as well. This work was

also supported by the projects ANR-12-BS01-0007 OPTIFORM ﬁnanced by

the French Agence Nationale de la Recherche (ANR).

1. Introduction

We study local minimizers of singular, discontinuous functionals of the

form

J(u, Ω) = ZΩω(x)|∇u|2+χ{u>0}dx −→ min,(1.1)

2010 Mathematics Subject Classiﬁcation. Primary 35B65. Secondary 35J60, 35J70.

Key words and phrases. Free boundary problems, Muckenhoupt weights, geometric

regularity.

1

2 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

where Ω is a bounded domain of Rd,d≥2, and ω(x) is a measurable,

singular A2weight in the sense that 0 ≤ω≤+∞, both ωand ω−1are

locally integrable, and

ZBr(z0)

ω! ZBr(z0)

ω−1!≤Cr2d,

for all balls Br(z0)⊂Ω. Thus, ωmay become zero or inﬁnity along a

lower-dimensional subset of Ω, hereafter denoted by:

Λ0(ω) := ω−1(0),Λ∞(ω) := ω−1(+∞).

If will also be convenient to denote Λ(ω) := Λ0(ω)∪Λ∞(ω). The class of

A2weight functions was introduced by Muckenhoupt [13] and is of central

importance in modern harmonic analysis and its applications. A canonical

example of an A2function is |x|αwith −d < α < d — having an isolated

singularity at the origin. Recent results, see for instance [3], show A2weights

play an important role in the theory of non-local diﬀusive problems. In

particular, when ω(x) = |x1|β,−1< β < 1, and the axis {x1= 0}is a

subset of ∂Ω, then (1.1) falls into the case studied in [2], where fractional

cavitation problems are considered.

The mathematical analysis of free boundary cavitation problems goes

back to the pioneering work Alt and Caﬀarelli [1], corresponding to the

case ω≡1 in (1.1). In the present work, we start the investigation of free

boundary problems of the type (1.1), for possibly singular weights, that is

Λ∞(ω)6=∅. Similarly, we can treat the degenerate case, i.e. when Λ0(ω)6=

∅. However, for didactical purposes, in this paper we shall restrict the

analysis to singular weights.

We show existence of a local minimizer u, and analyze analytic and weak

geometric properties of the free boundary ∂{u > 0} ∩Ω. The latter task is,

in principle, a delicate issue. For instance, one notices, for singular weights,

the existence of two distinct types of free boundary points:

(1) ∂{u > 0} ∩ Λ∞=∅;

(2) ∂{u > 0} ∩ Λ∞6=∅.

Case (1) refers to a non-homogeneous version of the Alt-Caﬀarelli prob-

lem, [1], as, away from the singular set, the weight is uniformly elliptic, see

for instance [4, 5, 9, 15]. Case (2) is rather more delicate, and in particular,

a central result we prove in this current work classiﬁes the geometric behav-

ior of a local minimizer near a free boundary point z0∈∂{u > 0} ∩ Λ∞,

in terms of its singularity rate near z0— a pure analytic information of

the problem. Indeed, we show local minimizers are precisely C1+γsmooth

along their corresponding free boundaries, where γis half of the geometric

blow-up rate of ωas it approaches the singular set Λ∞; see condition (H3)

for precise deﬁnitions.

SINGULAR FREE BOUNDARY PROBLEMS 3

The paper is organized as follows. In Section 2 we formally present the

minimization problem we shall study. A brief description of the initial math-

ematical tools required in the investigation of A2-singular free boundary

problems is also delivered in that section. In Section 3 we discuss existence

and L∞bounds for minimizers, whereas in Section 4 we establish various

compactness properties for family of minimizers. In Section 5 we prove the

key novel result of the paper, namely that solutions to A2cavitation prob-

lems are C1+γregular at their singular free boundary points, where γis

a sharp prescribe value. In particular, if z0is a free boundary point and

ω(z0)<+∞, that is it is non-singular, then γ(z0) = 0 and we recover the

classical Alt-Caﬀarelli Lipschitz regularity estimate for cavitation problems.

Finally in Section 6 we obtain a quantitative non-degeneracy estimate for

solutions near its singular free boundary points.

2. Mathematical set-up

In this section we give a precise description of the minimization problem

considered in this article and gather some of the main known results about

elliptic equations involving A2weights, required in our study.

Given an open set Ω ⊂Rd, we denote by M(Ω) the set of all real-valued

measurable functions deﬁned on Ω. A nonnegative locally integrable func-

tion ω: Ω →Ris said to be an A2weight if ω−1is also locally integrable

and

sup

B⊂Ω1

|B|ZB

ω 1

|B|ZB

ω−1≤C1,(2.1)

holds for a constant C1>0 and any ball B⊂Ω. Two weights are said to

belong to the same A2class if condition (2.1) is veriﬁed for both functions

with the same constant C1.

Fixed an A2weight ωand 1 ≤p < ∞, we deﬁne

Lp(Ω, ω) = (f∈ M(Ω)kfkLp(Ω,ω):= ZΩ|f(x)|pω(x)dx1/p

<∞).

Accordingly, we deﬁne the weighted Sobolev space as

W1,p(Ω, ω) := u∈Lp(Ω, ω)Diu∈Lp(Ω, ω),for i= 1,2,···d,

and, by convention, we write W1,2(Ω, ω) as H1(Ω, ω).

An A2weight ωgives raise to the degenerate/singular elliptic operator

Lω(·) = div(ω∇·),

which acts on H1(Ω, ω). In a series of three papers, [6, 7, 8], Fabes, Jerison,

Kenig, and Serapioni developed a systematic theory for this class of opera-

tors: existence of weak solutions, Sobolev embeddings, Poincar´e inequality,

Harnack inequality, local solvability in H¨older spaces, and estimates on the

4 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

Green’s function. Below we state four results from [6], supporting, directly

or indirectly, the framework developed in this present article.

Theorem 1 (Solvability in Sobolev spaces).Let Ω⊂Rdbe a smooth

bounded domain, h= (h1, ..., hn)satisfy |h|/ω ∈L2(Ω, ω), and g∈H1(Ω, ω ).

Then, there exists a unique solution u∈H1(Ω, ω)of Lωu=−div hin Ω

with u−g∈H1

0(Ω, ω).

Theorem 2 (Local H¨older regularity).Let Ω⊂Rdbe a smooth bounded

domain and ua solution of Lωu=−div hin Ω, where |h|/ω ∈L2(Ω, ω).

Then, uis H¨older continuous in Ωwith a H¨older exponent depending only

on dand the A2class of ω.

Theorem 3 (Harnack inequality).Let ube a positive solution of Lωu= 0

in B4R(x0)⊂Rd. Then, supBR(x0)u≤Cinf BR(x0)ufor some constant C

depending only on dand the A2class of ω— and in particular, independent

of R.

Theorem 4 (Poincar´e inequality).There is as positive constant Csuch that

for all Lipschitz continuous function udeﬁned on BR, the following holds

1

ω(BR)ZBR|u−AR|2ω≤CR1

ω(BR)ZBR|∇u|2ω

where ω(BR) = RBRωand ARis either 1

ω(BR)RBRu(x)ω(x)dx or RBRu(x)dx.

Let us now turn to the mathematical description of the problem to be

studied in the present article. Given a nonnegative boundary datum f∈

H1(Ω, ω)∩L∞(Ω), we consider the minimization problem

J(ω, u, Ω) = ZΩω(x)|∇u|2+χ{u>0}dx −→ min,(2.2)

among functions u∈H1

f(Ω, ω) := f+H1

0(Ω, ω), where χOstands for the

characteristic function of the set Oand ωis an A2weight.

We are mostly interested in local geometric properties of local minima

near a singular free boundary point. Henceforth, as to properly carry out

the analysis, it is convenient to localize the problem into the unit ball B1and

assume the origin is a free boundary point, i.e., 0 ∈∂{u > 0}.In addition

we shall assume throughout the paper the following structural condition on

the weight ω:

(H1) The weight ωbelongs to A2, and 0 < τ0≤ω≤+∞a.e. in Ω.

It turns out that (H1) prescribes minimal condition under which one

can develop an existence and regularity theory for corresponding singular

cavitation problem.

We comment that we have chosen to develop the analysis of problems

involving singular weights, rather than degenerate ones. In turn, we consider

strictly positive weights that may blow-up along its singular set Λ∞(ω). We

SINGULAR FREE BOUNDARY PROBLEMS 5

could similarly treat bounded, degenerate weights of order &rσ, for some

σ < 2.

A number of physical free boundary problems fall into the above mathe-

matical set-up. Typical examples of weight functions we have in mind are

ω(x) = |x′|α,

where x= (x′, xm+1,··· , xd), for 0 ≤m < d and −m < α ≤0. More

generally, if Nis an m-dimensional manifold properly embedded in Rd,

0≤m < d, we are interested in weights of the form

ω(x) = dist(x, N)α,

for some −m < α ≤0. This class of weight functions gives raise to the

analysis of free boundary problems ruled by diﬀusion operators with an

m-dimensional singular set. Anisotropic weights of the form

ω(x) :=

d

Y

i=1 |xi|αi,

also fall under the hypothesis considered in this work. In this case, condition

(H1) is veriﬁed as long as −1< αi≤0.

We are further interested in weights with possible distinct behaviors along

sets of diﬀerent dimensions, say

ω(x) = |(x1·x2···xm)|α1· |(xm+1 ···xd)|α2,

where −m < α1≤0, −d+m < α2≤0. In this model, 0 is a singular

free boundary point of degree |α1+α2|, as we shall term later. If we label

the cones C1:= {Πm

i=1xi= 0}and C2:= Πd

i=m+1xi= 0, then any free

boundary point in C1\C2is singular of degree |α1|and similarly, a point

in ∂{u > 0} ∩ (C2\C1) is singular of degree |α2|. Any other free boundary

point z∈∂{u > 0} \ (C1∪C2) is a free boundary point of degree 0.

Notice that no continuity assumption has been required on ω. In partic-

ular, we are interested in the analysis of free boundary problems in possibly

random media. Given a measurable function 0 < c0≤θ < c−1

0deﬁned on

the unit sphere Sd−1, 0 ≤m < d −2 and −m < α ≤0, we can always deﬁne

an α-homogeneous, A2weight function ωin B1\{0}as

ω(x) := |x′|α·θx

|x|,

where, as above, x= (x′, xm+1,···, xd). This is another important example

of weights we have in mind as to motivate this work.

We conclude this section by setting a nomenclature convention: hereafter

any constant that depends only upon dimension, d, and ωwill be called

universal.

6 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

3. Existence and local boundedness

In this preliminary section, we discuss about existence and local bound-

edness of minimizers. We also comment on the Euler-Lagrange equation

satisﬁed by a local minimum. The proofs follow somewhat classical argu-

ments, thus we simply sketch them here for the sake of completeness.

Fixed a nonnegative boundary datum f∈H1(Ω, ω), one can consider a

minimizing sequence vj∈H1

f(Ω, ω). Clearly,

C > J(vj)≥ZΩ

ω(x)|∇vj|2dx.

Thus, up to a subsequence, vj→uweakly in H1(Ω, ω), for some function

u∈H1

f(Ω, ω). By compactness embedding, vj→uin L2(Ω, ω) and vj→u

a.e. in Ω. Passing to the limit as j→ ∞ (see [1, Section 1.3] to handle the

term χ{vj>0}), one concludes

J(u)≤lim inf

j→∞ J(vj) = min J.

This shows the existence of a minimizer. To verify that uis nonnegative,

one simply compares uwith u+in the minimization problem. Also, if the

boundary datum fis assumed to be in H1(Ω, ω )∩L∞(Ω), then for each

|t| ≪ 1, the function u+t(kfk∞−u)−competes with uin the minimization

problem. Standard computations yield {u > kfk∞}has measure zero, that

is, {0≤u≤ kfk∞}has total measure in Ω.

Now, if ϕis a nonnegative test function in C∞

0(Ω), then u+ϕcompetes

with uin the minimization problem. As {u+ϕ > 0} ⊃ {u > 0}, there holds,

−2ZΩ

ω(x)∇u· ∇ϕdx =J(u+ϕ)−J(u)−ZΩχ{u+ϕ>0}−χ{u>0}dx ≥0.

This shows div (ω(x)∇u) deﬁnes a nonnegative measure ν. If Bδ(x0)⊂ {u >

0}, then given a test function ϕ∈C∞

0(Bδ(x0)), for 0 <|t| ≪ 1, u+tϕ is

also positive in Bδ(x0), thus we conclude ν≡0 in the interior of {u > 0}.

Let us gather the information delivered above and state as a theorem for

future reference.

Theorem 5. Let ωbe any A2weight and f∈H1(Ω, ω)nonnegative. Then

there exists a minimizer u∈H1(Ω, ω)to problem (2.2) such that u=f

on ∂Ω, in the trace sense. Furthermore, uis nonnegative, kukL∞(Ω) ≤

kfkL∞(Ω), and there exists a non-negative Radon measure ν, supported on

its free boundary ∂{u > 0}, such that

div (ω(x)∇u) = ν

is veriﬁed in the distributional sense.

SINGULAR FREE BOUNDARY PROBLEMS 7

4. Compactness

In this section we discuss several compactness properties pertaining to

the problem. In particular, we will show that bounded local minima are

(universally) locally H¨older continuous, thus any family of bounded local

minima is pre-compact in the uniform convergence topology.

We start with a lemma reminiscent of Caccioppoli inequality.

Lemma 6. Let ube a bounded local minimizer of (2.2) in B1,y∈B1/2

and r < 1/12 be ﬁxed. Denote by m:= kukL∞(B1)and let hbe the unique

solution of div (ω(x)∇h) = 0 in B2r(y)

h=uon ∂B2r(y).

Then there exist universal constants 0< µ < 1and C > 0, such that

ZBr(y)

ω(x)|∇h|2dx ≤Cm2·rd−2+2µ.(4.1)

Proof. Following standard Caccioppoli type methods, one easily reaches

ZBr(y)

ω(x)|∇h|2≤C

r2ZB5r/4(y)

ω(x)u−ZB5r/4(y)

u2.

By (H1), along with Reverse H¨older inequality for A∞weights, see for in-

stance [14], we know ω∈Lqfor some q > 1. Applying H¨older inequality

ZB5r/4(y)

ω(x)u−ZB5r/4(y)

u2

≤ZB5r/4(y)

ωq(x)1/qZB5r /4(y)

(u−ZB5r/4(y)

u)2q/q−1(q−1)/q

≤CZB5r/4(y)

(u−ZB5r/4(y)

u)2q/q−1(q−1)/q

Invoking Campanato theory in the last previous line, and denoting by µthe

H¨older exponent of h(recall that his H¨older continuous by Theorem 2), one

has ZB5r/4(y)

ω(x)u−ZB5r/4(y)

u2≤Cm2rd+2µ,

hence the result.

Theorem 7 (Local C0,τ ).Let ube local minimizer of (2.2) in B1. There

exist universal constants 0< τ ≪1and C > 1such that u∈C0,τ (B1/2)

and

kukC0,τ (B1/2)≤CkukL∞(B1).

8 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

Proof. Let ube a local minimizer of (2.2) in B1,y∈B1/2and r < 1/12 be

ﬁxed. Let hbe as in Lemma 6. Since hcompetes with uin the minimization

problem, we can estimate

ZB2r(y)

ω(x)|∇u|2− |∇h|2dx ≤cdrd.(4.2)

From the PDE satisﬁed by h, we have

ZB2r(y)

ω(x)∇h· ∇(h−u)dx = 0,(4.3)

hence,

ZB2r(y)

ω(x)|∇u− ∇h|2dx =ZB2r(y)

ω(x)|∇u|2− |∇h|2dx. (4.4)

Combining (4.1), (4.2), (4.3), and (4.4), together with the triangle inequality,

yields

k∇ukL2(Br(y),ω)≤ k∇(u−h)kL2(B2r(y),ω)+k∇hkL2(Br(y),ω)

≤cd·rd

2+√Cm ·rd

2−1+µ.(4.5)

By H¨older inequality and (H4), we can further estimate,

ZBr(y)|∇u|dx ≤ ZBr(y)

ω−1(x)dx!1

2

· ZBr(y)

ω(x)|∇u|2dx!1

2

≤τ−1/2

0rd

2·cd·rd

2+√Cm ·rd

2−1+µ

≤C4rd−1+µ,

(4.6)

for a constant C4>0 depending only on universal parameters. Local H¨older

continuity of ufollows now by Morrey’s Theorem, see [12].

Homogenization. We now turn our attention to limiting free boundary

problems arising from homogenization. That is, hereafter we assume

(H2) Let 0 be a free boundary point. There exists −d < α ≤0 such that,

as λ→0+,

ωλ(x) := λ|α|ω(λx),

converges locally in L1to a weight ω0(x)∈A2.

The weight ω0is the homogenization limit, as it veriﬁes ω0(tx) = tαω0(x),

for all t > 0. Of course, should the original weight ωbe homogeneous, then

(H2) is immediately veriﬁed.

The examples discussed at the end of Section 2 are, essentially, α–homogeneous.

In addition to classical homogenization procedures, condition (H2) further

contemplates perturbations of those by terms g(x) = o(|x|−|α|) as |x| → 0,

and products by bounded, positive functions θ(x)∈VMO. So typical, we

have in mind weights of the form

ω(x) = θ(x)ω0(x) + g(x),

SINGULAR FREE BOUNDARY PROBLEMS 9

where ω0is α-homogeneous (as in the examples from Section 2), θ∈VMO

verifying 0 < λ0< θ(x)< λ−1

0, and |x||α|g(x)→0 as |x| → 0.

Before we continue, let us make a comment on the scaling of the free

boundary problem (2.2), which further substantiates (H2).

Remark 8.Let ube a local minimizer of

J(ω, u, Ω) := ZΩ

ω(x)|∇u(x)|2+χ{u>0}dx.

Given 0 < λ < 1, deﬁne

β= 1 −α

2, uλ(x) = λ−βu(λx),and ωλ(x) = λ|α|ω(λx),

then change of variables yields

J(ω, u, Ω) = ZΩ/λ ω(λy)|∇u(λy)|2+χ{u(λy)>0}λddy

=ZΩ/λ hλαωλ(y)λ2(−1+β)|∇uλ(y)|2+χ{uλ(y)>0}iλddy

=λdJ(ωλ, uλ,Ω/λ).

That is, uλis a local minimizer of functional Jλ, ruled by an approximation

of the homogenizing medium.

We start oﬀ with a weak compactness result.

Proposition 9 (Weak compactness in W1,d+).Let λkbe any sequence con-

verging to zero and uklocal minima of Jk=J(ωk, v, Ω), with ωk(x) :=

λ|α|

kω(λkx). There exists µ > 0such that {∇uk}k≥1is locally weakly pre-

compact in Ld+µ(Ω).

Proof. We revisit the proof of Theorem 7, for ω=ωλk,u=ukas to reach

the corresponding of (4.6), namely

ZBr(y)|∇uk|dx ≤Crd−1+τ,(4.7)

for positive constants Cand τ > 0 independent of k. Fixed Ω′⋐Ω, by

standard applications of H¨older inequality, we can bound {|∇uk|}k∈Nin

Ld

1−τ(Ω′), uniformly in k. We choose µ > 0 such that d(1 −τ)−1=d+µ,

and the result follows since Ld+µis a reﬂexive space.

In the sequel we indeed show that local minima of the functional Jλcon-

verge to a minimizer of the singular homogenized problem ruled by ω0.

Theorem 10. Assume 0 is a free boundary point and that ωsatisﬁes H1-

H2. Let λkbe any sequence converging to zero and uklocal minima of

Jk=J(ωk, v, Ω), with ωk(x) := λ|α|

kω(λkx). Then, up to a subsequence, uk

converges locally uniformly to a local minimum of J(ω0, v, Ω).

10 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

Proof. From Theorem 7 along with Proposition 9, up to a subsequence, uk→

ulocally uniformly in Ω and locally weakly in W1,d+µ. By minimization,

ω1/2

k|∇uk|is bounded in L2. Also, from (H2), ω1/2

k→ω1/2

0strongly in L2,

and thus

ω1/2

k∇uk⇀ ω1/2

0∇u,

locally in the weak topology of L2. Lower weak semicontinuity of norm along

with Fatou’s Lemma imply, for any ﬁxed subdomain Ω′⋐Ω, there holds

J(ω0, u0,Ω′)≤lim inf J(ωk, uk,Ω′).(4.8)

Now, ﬁx a ball B:= Br(x0)⋐Ω and let ϕbe a smooth function in B

satisfying ϕ=u0on ∂B. For ε > 0 small, consider

ψ(y) := |y−x0| − r

εr

and deﬁne ϕε

k: Ω →Rto be the linear interpolation between ukand ϕ

within B(1+ε)r(x0), that is:

ϕε

k(y) =

ϕ(y),if |y−x0| ≤ r

uk(y),if |y−x0| ≥ (1 + ε)r

(1 −ψ(y))u0(y) + ψ(y)uk(y),if r < |y−x0|<(1 + ε)r.

(4.9)

Since ukis a local minimizer of Jkover e

B:= B(1+ε)r(x0), we have

J(ωk, uk,e

B)≤J(ωk, ϕε

k,e

B),

that is

J(ωk, uk,e

B)≤J(ωk, ϕ, B) + J(ωk, ϕε

k,e

B\B).(4.10)

On the transition region, r < |y−x0|<(1 + ε)r, taking into account (4.9),

we estimate for a.e. y,

|∇ϕε

k(y)| ≤ |∇u0(y)|+|∇uk(y)|+C

ε|uk(y)−u0(y)|.(4.11)

Consequently, from (4.9), (4.10) and (4.11), taking into account that |e

B\

B| ∼ εand that ωk(y)|∇uk|2is bounded in L1+δ(e

B), for some δ > 0, we

obtain, for some 0 < β ≪1,

J(ωk, uk,e

B)≤J(ωk, ϕ, B) + C εβ+ε−2+βo(1),(4.12)

as k→ ∞. Finally, letting k→ ∞, taking into account (4.8), we obtain

J(ω0, u0, B)≤J(ω0, ϕ, B ) + Cεβ.

Since ϕand ε > 0 were taken arbitrary, we conclude the proof of the Theo-

rem.

SINGULAR FREE BOUNDARY PROBLEMS 11

5. C1+γregularity at the free boundary

In the previous section we showed minimizers are locally H¨older contin-

uous, which, in particular, yields a rough oscillation control of unear the

free boundary. The heart of the matter, though, is to describe the precise

geometric behavior of a local minimizer at free boundary points. Roughly

speaking, solutions to singular free boundary problems should adjust their

vanishing rate based upon the singularity of the medium. More precisely, in

this section we shall work under the following assumption:

(H3) For some −d < α ≤0, there exists L > 0 such that

ZBr(0)

ω(x)dx ≤Lrα.

Recall, for notation convenience, we have localized 0 as a free boundary

point, thus, condition (H3) prescribes, in a way, the maximum blow-up

rate of ωat a free boundary. In this case, we will say 0 is a singular free

boundary point of degree |α|. In particular, regular free boundary points

from classical study of cavitation problems, that is such that ω(0) is ﬁnite,

represent (singular) free boundary points of degree zero.

While there is no hope to obtain an estimate superior than H¨older con-

tinuity of local minima, at any other point, in this section we show that

ubehaves as a C1+γfunction around a singular free boundary point. As

usual, this implies higher order diﬀerentiability of uat free boundary points.

In particular, if 1 + γ=Nis an integer, then u∈CN−1,1, in the sense it

is N−1 diﬀerentiable, and the Nth-Newtonian quotient remains bounded.

When 1 + γ=N+θ, for 0 < θ < 1, then solutions are CN,θ regular at free

boundary points.

This is the contents of the following key result:

Theorem 11. Assume (H1) −(H2) −(H3) and let ube a local minimizer

of (2.2) so that 0 is a free boundary point. Then

sup

Br

u≤Cr1+ |α|

2,

for a universal constant C > 0, independent of u.

The proof of Theorem 11 will be based on a geometric ﬂatness improve-

ment technic. For that, we need a Lemma:

Lemma 12. Let u∈H1(B1, ω)verify 0≤u≤1in B1, with u(0) = 0.

Given δ > 0, there exists ε > 0depending only on δand universal constants,

such that if uis a local minimizer of

Jε(u) := ZB1

ωλ(x)|∇u(x)|2+εχ{u>0}dx, (5.1)

12 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

for any 0< λ < 1, then, in B1/2,uis at most δ, that is,

sup

B1/2

u≤δ. (5.2)

Proof. Suppose, for the sake of contradiction, that the thesis of the Lemma

does not hold. This means for some δ0>0, one can ﬁnd a sequence of

functions uj∈H1(B1, ωj) satisfying:

(1) 0 ≤uj≤1 in B1;

(2) uj(0) = 0;

(3) ujis a local minimizer of Jj(u) := RB1ωλj(x)|∇u(x)|2+1

jχ{u>0}dx;

however,

sup

B1/2

uj≥δ0∀j∈N.(5.3)

From compactness estimates proven in Section 4, up to a subsequence, uj

converges locally uniformly to a function a nonnegative function u∞, with

u∞(0) = 0. Also, from similar analysis as in Theorem 10, u∞is a local

minimizer of

J0(v) := ZB1

ω0(x)|∇v(x)|2dx.

Applying maximum principle (available for minimizers of the functional J0),

we conclude u∞≡0. That is, we have proven ujconverges locally uniformly

to 0. Therefore, for j≫1, we reach a contradiction with (5.3). The proof

of Lemma 12 is complete.

Proof of Theorem 11. We will make few (universal) decisions. Initially we

set

δ⋆:= 2α

2−1.

Lemma 7 assumes the existence of a positive (universal) constant ε0>0

such that any normalized minimizer of Jε0, as deﬁned in (5.1) veriﬁes (5.2),

for δ⋆. Deﬁne

0:= 2−α

√ε0,and ˜u(x) := u(0x).

By remark 8, ˜uis a local minimizer of Jε0, and hence, from Lemma 7, there

holds:

sup

B1/2

˜u(x)≤2α

2−1.(5.4)

Next, by induction, we iterate the previous argument as to show

sup

B2−k

˜u(x)≤2k(α

2−1).(5.5)

Estimate (5.4) gives the ﬁrst step of induction, k= 1. Now, suppose we

have veriﬁed (5.5) for k= 1,2,···p. Deﬁne

˜v(x) := 2p(1−α

2)·˜u2−px.

SINGULAR FREE BOUNDARY PROBLEMS 13

It follows from induction hypothesis that 0 ≤v≤1. Also, from scaling, we

check that ˜vis too a minimizer of Jε0. Applying Lemma 7 to ˜vwe conclude

sup

B1/2

˜v≤2α

2−1,

which, in terms of ˜u, gives the precisely the (p+ 1) step of induction.

Now, given a (universally small) radius r > 0, choose k∈N, such that

2−(k+1) < −1

0r≤2−k.

We can then estimate

sup

Br

u= sup

B−1

0r

˜u

≤sup

B2−k

˜u

≤2k(α

2−1)

≤2

01−α

2

·r1+ |α|

2

=Cr1+ |α|

2,

for C > 1 universal, as required.

6. Nondegeneracy and weak geometry

In the previous section we show local minima are C1+γsmooth along the

singular free boundary, for γ=|α|

2. In this section we prove a competing

inequality which assures that such a geometric decay is sharp. Naturally,

such an estimate requires a corresponding lower bound for the degree of

singularity of the free boundary, namely:

(H4) For some −d < α ≤0, there exists τ⋆>0 such that

τ⋆rα≤ZBr(0)

ω(x)dx,

for all 0 < r ≪1.

Again, we recall 0 is the localized free boundary point we are analyz-

ing, thus condition (H4) conveys the idea that the origin is a singular free

boundary point of degree at least |α|. Note that (H4) yields

inf

r>0ZB1

ωr(x)dx ≥τ⋆,

where, as before, ωr(x) = r|α|ω(rx).

Theorem 13. Let ube a minimizer of f (2.2),0a free boundary point and

assume (H4). Then

sup

Br

u(x)≥2·sτ⋆

dd

(d+ 2)d+2 ·r1+ |α|

2,∀0< r < 1,

14 J. LAMBOLEY, Y. SIRE, AND E.V. TEIXEIRA

where dis dimension.

Proof. The idea of the proof is to cut a family of concentric holes on the

graph of u, compare the resulting functions with uin terms of the mini-

mization problem J, and ﬁnally optimize the cutting-hole parameter; here

are the details. Let 0 < r < 1 be a ﬁxed radius and deﬁne vr:B1→Ras

vr(y) := rα

2−1u(ry).

The goal is to show that

Sr:= sup

B1

vr

is uniformly bounded from below, independently of r. From Remark 8, vr

is a local minimizer of Jrover B1. Next, let us choose 0 < σ < 1, 0 < ε ≪1

and craft a smooth, radially symmetric function ϕ:B1→Rsatisfying:

0≤ϕ≤1, ϕ ≡0 in Bσ, ϕ = 1 on ∂B1,|∇ϕ| ≤ (1+ε)(1−σ)−1.(6.1)

In the sequel, let us consider the test function ξ:B1→Rgiven by

ξ(x) := min {v(x),(1 + ε)Sr·ϕ(x)}.

By construction, ξcompetes with vin the minimization problem Jr, and

thus

ZB1ωr(x)|∇ξ|2+χ{ξ>0}dx ≥ZB1ωr(x)|∇v|2+χ{v>0}dx. (6.2)

We can rewrite (6.2) as an inequality of the form A ≥ B, for

A:= ZB1

ωr(x)|∇ξ|2−|∇v|2dx,

B:= ZB1χ{v>0}−χ{ξ>0}dx.

(6.3)

As to estimate Bfrom below we note that {v > 0} ⊃ {ξ > 0}, thus

B=ZB1

χ{ξ=0}dx ≥Ln(Bσ).(6.4)

Next we estimate Afrom above. For that, let us deﬁne

Π := x∈B1(1 + ε)Sr·ϕ(x)< v(x)

SINGULAR FREE BOUNDARY PROBLEMS 15

and compute

ZB1

ωr(y)|∇ξ|2− |∇v|2dx =Z

Π

ωr(x)|∇ξ|2− |∇v|2dx

≤(1 + ε)2S2

rZ

B1

ωr(x)|∇ϕ|2dx

≤(1 + ε)4(1 −σ)−2·

Z

B1

ωr(x)dx

·S2

r.

(6.5)

From the relation A ≥ B, we obtain

S2

r≥τ⋆

(1 + ε)4σd(1 −σ)2.

Letting ε→0 and selecting σ=d

d+ 2 yields the optimal lower bound. The

proof of Theorem 13 is complete.

A important consequence of Theorem 11 and Theorem 13 combined is

that, around free boundary points of same homogeneity α, a local minimum

detaches from its coincidence set, {u= 0}, precisely as dist1+ |α|

2.

Corollary 14. Let ube a minimizer of (2.2) and assume all free boundary

points in B1/2satisfy conditions (H1)—(H4). Then there exists a universal

constant C > 1such that

C−1dist(x, ∂ {u > 0})1+ |α|

2≤u(x)≤C·dist(x, ∂ {u > 0})1+ |α|

2

for any x∈B1/4∩ {u > 0}.

By standard arguments we then conclude that near free boundary points

of same homogeneity, the set of positivity {u > 0}has uniform positive

density:

Corollary 15. Let ube a minimizer of (2.2) and assume all free boundary

points in B1/2satisfy (H1)—(H4) Then, for any 0< r < 1

4,

Ln(Br∩ {u > 0})

Ln(Br)≥µ,

where 0< µ < 1is a universal constant, independent of r.

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Sorbonne Universit´

es, UPMC Univ Paris 06, Institut de Math´

ematiques de

Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne

Paris Cit´

e, F-75005, Paris, France

E-mail address:jimmy.lamboley@imj-prg.fr

Johns Hopkins University, Krieger Hall, N. Charles St., Baltimore, MD

21218

E-mail address:sire@math.jhu.edu

University of Central Florida, 4393 Andromeda Loop N, Orlando, FL 32816

E-mail address:eduardo.teixeira@ucf.edu