Two-Phase Free Boundary Problems: From Existence to Smoothness

Daniela De Silva; Fausto Ferrari; Sandro Salsa

doi:10.1515/ans-2017-0015

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Two-Phase Free Boundary Problems: From Existence to Smoothness

Daniela De Silva , Fausto Ferrari and Sandro Salsa

Published/Copyright: April 21, 2017

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From the journal Advanced Nonlinear Studies Volume 17 Issue 2

Abstract

We describe the theory we developed in recent times concerning two-phasefree boundary problems governed by elliptic operators with forcing terms.Our results range from existence of viscosity solutions to smoothness ofboth solutions and free boundaries. We also discuss some open questions,possible object of future investigation.

Keywords: Two-Phase Free Boundary Problem; Free Boundary Regularity; Perron Solutions, Elliptic Systems

MSC 2010: 35R65; 35J57; 35B65

1 Introduction

In the last few years, significant progress has been achieved in theanalysis of two-phase free boundary problems governed by ellipticequations with forcing terms. In this survey paper we give an account of ourmain contributions and to avoid technicalities we refer to the followingmodel problem:

(1.1) { Δ ⁢ u = f + in ⁢ B 1 + ⁢ ( u ) , Δ ⁢ u = f - in ⁢ B 1 - ⁢ ( u ) , | ∇ ⁡ u + | 2 - | ∇ ⁡ u - | 2 = 1 on ⁢ F ⁢ ( u ) := ∂ ⁡ B 1 + .

Here B1 is the unit ball in ℝn, centered at the origin, f±∈C⁢(B1)∩L∞⁢(B1) and

B 1 + ⁢ ( u ) := { x ∈ B 1 : u ⁢ ( x ) > 0 } , B 1 - ⁢ ( u ) := { x ∈ B 1 : u ⁢ ( x ) ≤ 0 } ∘ .

Moreover, u+ and u- denote the positive and negative part of u,respectively. F⁢(u) is the so-called free boundary of u.

This type of problem arises in a number of applied contexts, such as thePrandtl–Bachelor model in fluid-dynamics (see, e.g., [2, 15]), theeigenvalue problem in magnetohydrodynamics [21], or in flamepropagation models [24]. We will comment on more general problems inthe final section of this paper.

The theory for problem (1.1) can be developed according to a well-established paradigm:

Existence and optimal regularity of solutions, e.g., viscosity orvariational solutions, or solutions obtained as a limit of singularperturbations.
Weak regularity properties of the free boundary, such as finite perimeterand density properties for the positivity set.
Strong regularity properties of the free boundary. For instance Lipschitzor “flat” free boundaries are C1 or better.
Higher regularity: Schauder estimates and analyticity for bothsolution and free boundary.

In the homogeneous case, i.e. f±=0, in his pioneer work [3, 5]Caffarelli obtained strong regularity properties of the free boundary. Subsequently,in [4] he showed existence of Lipschitz viscosity solutions whichenjoy weak regularity properties of the free boundary. The Lipschitz regularity ofviscosity solutions to homogeneous problems relies on a monotonicity formulaby Alt, Caffarelli and Friedman [1]. For the inhomogeneous case,Lipschitz regularity was obtained by Caffarelli, Jerison and Kenig in [6]. For further results on homogeneous free boundary problem see forexample [18, 17, 19, 20, 28, 29].

The paper is organized as follows. In Sections 2 and 3 we provide a briefdescription of the results and the main ideas introduced in the papers [8, 9, 11, 12],concerning points (a), (b) and (c) above. We focus mostly on the strong regularity results, as our novel approach differs from the above mentioned work of Caffarelli. Section 4 focuses onpoint (d), i.e. higher regularity issues with an account of our recentresults in [13]. Our higher regularity results for two-phase problemsare new even in the homogeneous case, with the exception of [16], wherea free boundary problem for the harmonic measure is considered.Interestingly, the proof of higher regularity presents somewhat unexpectedfeatures, proper of genuine two-phase problems. In Section 5 weindicate possible generalization of our results and emphasize some open questions.

2 Existence of Lipschitz Viscosity Solutions and Weak Regularity Properties of the Free Boundary

In this section we describe our existence result. In [11], we usePerron’s method to construct Lipschitz viscosity solutions tofree boundary problems withforcing terms (for a given boundary data), thus extending the results of[4] to the inhomogeneous case. Our results hold for operators ℒ=div⁡(A⁢(x)⁢∇) with Hölder coefficientsand general free boundary conditions |∇⁡u+|=G⁢(|∇⁡u-|),with G Lipschitzcontinuous, strictly increasing and G⁢(0)>0. In this sectionhowever we consider the model problem (1.1).

We start by recalling the definition of viscosity solution.

We say that a point x0∈F⁢(u) is regular from the right (resp. left)if there is a ball B⊂B1+⁢(u) (resp. B1-⁢(u)),such that B∩F⁢(u)={x0}.

Definition 2.1

We say that u∈C⁢(B1) is a viscosity solution of free boundary problem (1.1)if the following holds:

Δ ⁢ u = f + in B1+⁢(u) and Δ⁢u=f- in B1-⁢(u).
u satisfies the free boundary condition in the following sense:
1. If x0∈F⁢(u) is regular from the right with tangent ball Bthen
  
  u + ⁢ ( x ) ≥ α ⁢ 〈 x - x 0 , ν 〉 + + o ⁢ ( | x - x 0 | ) ⁢ in ⁢ B , with ⁢ α ≥ 0 ,
  
  u - ⁢ ( x ) ≤ β ⁢ 〈 x - x 0 , ν 〉 - + o ⁢ ( | x - x 0 | ) ⁢ in ⁢ B c , with ⁢ β ≥ 0
  
  with equality along every non-tangential domain, and α2-β2≤1.
2. If x0∈F⁢(u) is regular from the left with tangent ball B,then
  
  u - ⁢ ( x ) ≥ β ⁢ 〈 x - x 0 , ν 〉 + + o ⁢ ( | x - x 0 | ) ⁢ in ⁢ B , with ⁢ β ≥ 0 ,
  
  u + ⁢ ( x ) ≤ α ⁢ 〈 x - x 0 , ν 〉 - + o ⁢ ( | x - x 0 | ) ⁢ in ⁢ B c , with ⁢ α ≥ 0
  
  with equality along every non-tangential domain, and α2-β2≥1.
Here ν=ν⁢(x0) denotes the unit normal to ∂⁡Bat x0, pointing towards B1+⁢(u).

The notion of viscosity solution can be also given in terms of testfunctions. Given u,φ∈C⁢(B1), we say that φ touches uby below (resp. above) at x0∈B1, if u⁢(x0)=φ⁢(x0)and

u ( x ) ≥ φ ( x ) ( resp. u ( x ) ≤ φ ( x ) ) in a neighborhood O of x 0 .

Then u∈C⁢(B1) is a viscosity solution to (1.1) if(i) holds and (ii) is replaced by (ii’):

Let x0∈F⁢(u) and v∈C2⁢(B+⁢(v)¯)∩C2⁢(B-⁢(v)¯) (B=Bδ⁢(x0),δ small) with F⁢(v)∈C2. If v touches u by below (resp. above) at x0, then

| ∇ v + ( x 0 ) | 2 - | ∇ v - ( x 0 ) | 2 ≤ 1 ( resp. ≥ 1 ) .

Our solution is constructed as the infimum over a class of admissiblesupersolutions ℱ which we define below.

Definition 2.2

A function w∈C⁢(B¯1) is in ℱ if the following holds:

w is a solution to

{ Δ ⁢ w ≤ f + in ⁢ B 1 + ⁢ ( w ) , Δ ⁢ w ≤ f - ⁢ χ { w < 0 } in ⁢ B 1 - ⁢ ( w ) .
If x0∈F⁢(u) is regular from the left, then, near x0,

w + ≤ α ⁢ 〈 x - x 0 , ν ⁢ ( x 0 ) 〉 + + o ⁢ ( | x - x 0 | ) , α ≥ 0 ,

w - ≥ β ⁢ 〈 x - x 0 , ν ⁢ ( x 0 ) 〉 - + o ⁢ ( | x - x 0 | ) , β ≥ 0 ,

with α2-β2<1.
If x0∈F⁢(w) is not regular from the left, then, near x0,

w ⁢ ( x ) = o ⁢ ( | x - x 0 | ) .

We also need to introduce a minorant subsolution. We say that a locally Lipschitz function u¯, defined in B1, is a minorant if the following holds:

u ¯ is a weak solution to

{ Δ ⁢ u ¯ ≥ f + in ⁢ B 1 + ⁢ ( u ¯ ) , Δ ⁢ u ¯ ≥ f - ⁢ χ { u ¯ < 0 } in ⁢ B - ⁢ ( u ¯ ) .
Every x0∈F⁢(u) is regular from the right and, near x0,

u ¯ - ≤ β ⁢ 〈 x - x 0 , ν ⁢ ( x 0 ) 〉 + + o ⁢ ( | x - x 0 | ) ,

u ¯ + ≥ α ⁢ 〈 x - x 0 , ν ⁢ ( x 0 ) 〉 - + o ⁢ ( | x - x 0 | ) ,

with α2-β2>1.

We are now ready to present our main result. Consider the problem

(2.1) { Δ ⁢ u = f + in ⁢ B 1 + ⁢ ( u ) , Δ ⁢ u = f - ⁢ χ { u < 0 } in ⁢ B 1 - ⁢ ( u ) , | ∇ ⁡ u + | 2 - | ∇ ⁡ u - | 2 = 1 on ⁢ F ⁢ ( u ) := ∂ ⁡ B 1 + .

Theorem 2.3

Let φ be a continuous function on ∂⁡B1and u¯ a minorant of our free boundary problem, withboundary data φ. Then

u = inf ⁡ { w : w ∈ ℱ , w ≥ u ¯ ⁢ in ⁢ B 1 ¯ }

is a locally Lipschitz viscosity solution to (2.1) such that u=φon ∂⁡B1, as long as the set on the right is non-empty. The freeboundary F⁢(u) has finite (n-1)-dimensional Hausdorff measure and thereexist universal positive constants c,C,r0 such that for every r<r0and every x0∈F⁢(u),

c ⁢ r n - 1 ≤ ℋ n - 1 ⁢ ( F ⁢ ( u ) ∩ B r ⁢ ( x 0 ) ) ≤ C ⁢ r n - 1 .

Moreover, if F∗⁢(u) denotes the reduced part of F⁢(u), then

ℋ n - 1 ⁢ ( F ⁢ ( u ) ∖ F ∗ ⁢ ( u ) ) = 0 .

The proof follows the main guidelines of [4]. The presence of adistributed source requires to face some new technical delicate points. Forinstance, the classical harmonic replacement technique does not work in thiscontext and one has to resort to suitable auxiliary obstacle problems. Forthe details we refer to [11].

The following theorem is a consequence of the results in the next sections.

Theorem 2.4

F ⁢ ( u ) is a C1,γ¯ surface in a neighborhood ofHn-1 a.e. point x0∈F⁢(u). Moreover, if f±∈Ck,γ (resp. C∞, analytic) then F⁢(u) is Ck+2,γ*(resp. C∞, analytic) in a neighborhood of Hn-1 a.e.point x0∈F⁢(u), for a universal γ* depending on n,γ, ∥f±∥k,γ.

3 Strong Regularity Results

In this section we describe our strong regularity results for the free boundary.Precisely, we explain the strategy to show that flat or Lipschitzfree boundaries are C1,γ. Our strategy differsfrom the one developed in [3, 5] for the homogeneous case. For thedetails of the proofs we refer to [9].

A way to express the flatness of the free boundary is to assume that F⁢(u)(or the zero set of u+) is trapped between two parallel hyperplanes at δ-distance from each other, for a small δ (δ-flatness). While this looks like a somewhat strong assumption, it is indeed anatural one since it is satisfied for example by rescaling a solution arounda point of the free boundary where there is a normal in some weak sense (regular point), for instance in the measure theoretical one (see Theorem 2.4).

Our main theorems read as follows [9]. We assume that f± arecontinuous with ∥f±∥L∞⁢(B1)≤L and let ube a Lipschitz viscosity solution to (1.1) in B1, with Lip⁡(u)≤L. Universal constants depend only on n,L.

Theorem 3.1

There exists a universal constant δ¯>0 suchthat, if 0≤δ≤δ¯ and

(3.1) { x n ≤ - δ } ⊂ B 1 ∩ { u + ( x ) = 0 } ⊂ { x n ≤ δ } ( δ -flatness ) ,

then F⁢(u) is C1,γ in B1/2, with γ universal.

Theorem 3.2

If F⁢(u) is a Lipschitz graph in B1, then F⁢(u) is C1,γ in B1/2, with γ universal.

Theorem 3.2 follows from Theorem 3.1. Here is an outlineof the proof. First we show that we can find σ>0, small, dependingon u such that F⁢(u) is a C1,γ graph in Bσ. Indeed,there exists a blow-up sequence uk=u⁢(rk⁢x)/rk whichconverges as rk→0, up to rotations, to a two-plane solution

U β ⁢ ( x n ) = α ⁢ x n + - β ⁢ x n - , β ≥ 0 , α 2 - β 2 = 1 .

This follows from a Weiss-type monotonicity formula and a dimensionreduction argument. The conclusion now follows from the flatness Theorem 3.1.

Next we use a compactness argument to show that σ depends only onthe Lipschitz constant L of F⁢(u). For this we need to show that F⁢(u)is δ¯-flat in Br for some r≥σ depending on L.If by contradiction no such σ exists, then we can find a sequenceof solutions uk and of σk→0 such that F⁢(uk) isnot δ¯-flat in any Br with r≥σk. Then the uk converge uniformly (up to a subsequence) to a solution u∗and we reach a contradiction since F⁢(u∗) is C1,γ in aneighborhood of 0 by the first part of the proof.

3.1 Non-degenerate Versus Degenerate

The proof of Theorem 3.1 is based on an iterative procedure that“squeezes” our solution around an optimal limiting configuration Uβ⁢(x⋅ν) at a geometric rate in dyadically decreasingballs. Here ν is a unit vector, which plays the role of the normalvector at the origin (say 0∈F⁢(u)). This plan of flatness improvementworks nicely in the one-phase case (β=0) or as long as the two phasesu+,u- are, say, comparable (non-degenerate case). Thedifficulties arise when the negative phase becomes very small but at thesame time not negligible (degenerate case). In this case the flatnessassumption in Theorem 3.1 gives a control of the positive phaseonly, through the closeness to a one-plane solutionU0⁢(x⋅ν)=(x⋅ν)+.

First of all, the flatness condition (3.1) implies that u is closeto Uβ for some β≥0. Indeed we prove that, for smalluniversal parameters η and ρ¯,

(3.2) ∥ u - U β ∥ L ∞ ⁢ ( B ρ ¯ ) ≤ η ⁢ ρ ¯ .

A closer look to (3.2) reveals that, when α and β arecomparable, a nice control on the location of F⁢(u) isavailable. However, when β≪α, only a one-side control of F⁢(u) is possible. This dichotomy is made evident if wetranslate the “vertical” closenessbetween the graphs of u and Uβ, given by (3.2), into“horizontal” closeness. By rescaling wemay take ρ¯=1 in (3.2).Then, setting η1/3=ε, we get the following lemma.

Lemma 3.3

If β≥ε, then

U β ⁢ ( x n - ε ) ≤ u ⁢ ( x ) ≤ U β ⁢ ( x n + ε ) in ⁢ B 3 / 4 .

If β<ε, then

U 0 ⁢ ( x n - ε ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε ) in ⁢ B 3 / 4 .

Thus, the dichotomy non-degenerate versus degenerate translatesquantitatively into the two cases:

β ≥ ε : non-degenerate , β < ε : degenerate .

3.2 Improvement of Flatness. Non-degenerate Case

In this case, the basic step in the improvement of flatness reads asfollows. Assume that for some ε>0, small, we have

(3.3) U β ⁢ ( x n - ε ) ≤ u ⁢ ( x ) ≤ U β ⁢ ( x n + ε ) in ⁢ B 1 ,

with 0<β≤L, α2-β2=1, and say 0∈F⁢(u). Onewould like to get in a smaller ball an improvement of (3.3). After arescaling we may assume that f is small compared to β, in particular,

∥ f ∥ L ∞ ⁢ ( B 1 ) ≤ ε 3 ≤ ε 2 ⁢ min ⁡ { α , β } .

Lemma 3.4

If 0<r≤r0 for r0universal, and 0<ε≤ε0 for someε0 depending on r, then

(3.4) U β ′ ⁢ ( x ⋅ ν 1 - r ⁢ ε 2 ) ≤ u ⁢ ( x ) ≤ U β ′ ⁢ ( x ⋅ ν 1 + r ⁢ ε 2 ) in ⁢ B r ,

with |ν1|=1, |ν1-en|≤C~⁢ε, and |β-β′|≤C~⁢β⁢εfor a universal constant C~.

To prove Theorem 3.1 we rescale considering a blow-up sequence

u k ⁢ ( x ) = u ⁢ ( r ¯ k ⁢ x ) r ¯ k , x ∈ B 1

for suitable r¯, and iterate Lemma 3.4 to get, at the k-th step,

U β k ⁢ ( x ⋅ ν k - r ¯ k ⁢ ε k ) ≤ u ⁢ ( x ) ≤ U β k ⁢ ( x ⋅ ν k + r ¯ k ⁢ ε k ) in ⁢ B r ¯ k ,

with εk=2-k⁢ε, |νk|=1,|νk-νk-1|≤C~⁢εk-1, and

| β k - β k - 1 | ≤ C ~ ⁢ β k - 1 ⁢ ε k - 1 , ε k ≤ β k ≤ L .

Note that at each step we have the correct inductive hypotheses.

This implies that F⁢(u) is C1,γ at the origin.Repeating the procedure for points in a neighborhood of x=0, since allestimates are universal, we conclude that there exist a unit vector ν∞=lim⁡νk and C>0, γ∈(0,1], both universal,such that, in the coordinate system e1,…,en-1,ν∞, ν∞⊥ej, ej⋅ek=δj⁢k, F⁢(u)is a C1,γ graph, say xn=f⁢(x′), with f⁢(0′)=0 and

| f ⁢ ( x ′ ) - ν ∞ ⋅ x ′ | ≤ C ⁢ | x ′ | 1 + γ

in a neighborhood of x=0.

The main question is: Where is the information allowing one to realizethe step from (3.3) to (3.4) hidden? Here a linearized problem comes into play.

3.3 The Linearized Problem

The flatness condition (3.3) suggests the renormalization

u ~ ε ⁢ ( x ) = { u ⁢ ( x ) - α ⁢ x n α ⁢ ε , x ∈ B 1 + ⁢ ( u ) ∪ F ⁢ ( u ) , u ⁢ ( x ) - β ⁢ x n β ⁢ ε , x ∈ B 1 - ⁢ ( u )

(3.5) u ⁢ ( x ) = { α ⁢ x n + ε ⁢ α ⁢ u ~ ε ⁢ ( x ) , x ∈ B 1 + ⁢ ( u ) ∪ F ⁢ ( u ) , β ⁢ x n + ε ⁢ β ⁢ u ~ ε ⁢ ( x ) , x ∈ B 1 - ⁢ ( u ) .

In (3.5), u appears as a first-order perturbation of Uβ⁢(xn). The idea is that the key information we are looking for is storedprecisely in the “coefficient” u~ε. To extract it, we look at what happens to u~ε, asymptotically as ε→0. Note that,as ε→0, B1+(u)→{xn>0}, B1-(u)→{xn<0} andF⁢(u) goes to {xn=0}, all in Hausdorff distance. We have

Δ ⁢ u ~ ε = f + α ⁢ ε ∼ ε in ⁢ B 1 + ⁢ ( u ) ∪ B 1 - ⁢ ( u ) .

On F⁢(u),

| ∇ ⁡ u + | = α ⁢ | e n + ε ⁢ ∇ ⁡ u ~ ε + ⁢ ( x ) | ∼ α ⁢ ( 1 + ε ⁢ ( u ~ ε + ) x n + ε 2 ⁢ | ∇ ⁡ u ~ ε + | 2 ) ,

| ∇ ⁡ u - | = β ⁢ | e n + ε ⁢ ∇ ⁡ u ~ ε - ⁢ ( x ) | ∼ β ⁢ ( 1 + ε ⁢ ( u ~ ε - ) x n + ε 2 ⁢ | ∇ ⁡ u ~ ε - | 2 )

and

0 = | ∇ ⁡ u + | 2 - | ∇ ⁡ u - | 2 - 1 ∼ 2 ⁢ ε ⁢ [ α 2 ⁢ ( u ~ ε + ) x n - β 2 ⁢ ( u ~ ε - ) x n ] + O ⁢ ( ε 2 ) .

Dividing by ε and letting ε→0, we getfor “the limit” u~ of uε, the following problem:

(3.6) Δ ⁢ u ~ = 0 in ⁢ B 1 + ∪ B 1 -

and the transmission condition (linearization of the free boundary condition)

(3.7) α 2 u ~ x n + - β 2 u ~ x n - = 0 on B 1 ∩ { x n = 0 } .

The crucial information we were mentioning before is contained in thefollowing regularity result.

Theorem 3.5

Let u~ be a viscosity solution to the transmissionproblem (3.6)–(3.7) in B1 such that ∥u~∥∞≤1. Then u~∈C∞⁢(B¯1±) and in particular, there exists a universal constant C¯ such that

(3.8) | u ~ ⁢ ( x ) - u ~ ⁢ ( 0 ) - ( ∇ x ′ ⁡ u ~ ⁢ ( 0 ) ⋅ x ′ + p ~ ⁢ x n + - q ~ ⁢ x n - ) | ≤ C ¯ ⁢ r 2 in ⁢ B r

for all r≤12 and with α2⁢p~-β2⁢q~=0.

The question is now to transfer estimate (3.8) to u~ε and then read it in terms of flatness for u through theformulas (3.5). The right way to proceed is to argue bycontradiction.

Fix r≤r0, to be chosen suitably. Assume that for a sequence εk→0 there is a sequence uk of solutions ofour free boundary problem in B1, with right-hand side fk such that∥fk∥L∞⁢(B1)≤εk2⁢βkand

U β k ⁢ ( x n - ε k ) ≤ u k ⁢ ( x ) ≤ U β k ⁢ ( x n + ε k ) in ⁢ B 1 ⁢ 0 ∈ F ⁢ ( u k ) ,

with 0≤βk≤L, αk2-βk2=1, but theconclusion of Lemma 3.1 does not hold for every k≥1.

Construct the corresponding sequence of renormalized functions

(3.9) u ~ k ⁢ ( x ) = { u k ⁢ ( x ) - α k ⁢ x n α k ⁢ ε k , x ∈ B 1 + ⁢ ( u k ) ∪ F ⁢ ( u k ) , u k ⁢ ( x ) - β k ⁢ x n β k ⁢ ε k , x ∈ B 1 - ⁢ ( u k ) .

At this point we need a compactness property to show that uk convergesuniformly (up to a subsequence) to a limit function u~, Höldercontinuous in B1/2. Also αk→α, βk→β, with α2-β2=1. The compactness isprovided by the Harnack inequality stated in Theorem 3.6 and itscorollary, as we shall see later.

It turns out that the limit function u~ satisfies the linearizedproblem (3.6)–(3.7) in the viscosity sense. Hence, from (3.8), having u~⁢(0)=0,

(3.10) | u ~ ⁢ ( x ) - ( x ′ ⋅ ν ′ + p ~ ⁢ x n + - q ~ ⁢ x n - ) | ≤ C ⁢ r 2 , x ∈ B r ,

for all r≤14 (say), with

α 2 ⁢ p ~ - β 2 ⁢ q ~ = 0 , | ν ′ | = | ∇ x ′ ⁡ u ~ ⁢ ( 0 ) | ≤ C .

Since u~k converges uniformly to u~ in B1/2,inequality (3.10) transfers to u~k:

| u ~ k ⁢ ( x ) - ( x ′ ⋅ ν ′ + p ~ ⁢ x n + - q ~ ⁢ x n - ) | ≤ C ′ ⁢ r 2 , x ∈ B r .

Set

β k ′ = β k ⁢ ( 1 + ε k ⁢ q ~ ) , ν k = 1 1 + ε k 2 ⁢ | ν ′ | 2 ⁢ ( e n + ε k ⁢ ( ν ′ , 0 ) ) .

Then,

α k ′ = 1 + β k ′ ⁣ 2 = α k ⁢ ( 1 + ε k ⁢ p ~ ) + O ⁢ ( ε k 2 ) , ν k = e n + ε k ⁢ ( ν ′ , 0 ) + ε k 2 ⁢ τ , | τ | ≤ C ,

where to obtain the first equality we used that α2⁢p~-β2⁢q~=0 and hence

β k 2 α k 2 ⁢ q ~ = p ~ + o ⁢ ( 1 ) .

With these choices we can now show that (for k large and r≤r0)

U β k ′ ⁢ ( x ⋅ ν k - ε k ⁢ r 2 ) ≤ u k ⁢ ( x ) ≤ U β k ′ ⁢ ( x ⋅ ν k + ε k ⁢ r 2 ) in ⁢ B r

leading to a contradiction.

We are left to prove the compactness claim. The Harnack inequality takes thefollowing form.

Theorem 3.6

There exists a universal ε~>0 such that,if x0∈B1 and u satisfies the condition

(3.11) U β ⁢ ( x n + a 0 ) ≤ u ⁢ ( x ) ≤ U β ⁢ ( x n + b 0 ) in ⁢ B r ⁢ ( x 0 ) ⊂ B 1

with

∥ f ∥ L ∞ ⁢ ( B 2 ) ≤ ε 2 ⁢ β , 0 < β ≤ L

and

0 < b 0 - a 0 ≤ ε ⁢ r

for some 0<ε≤ε~, then

U β ⁢ ( x n + a 1 ) ≤ u ⁢ ( x ) ≤ U β ⁢ ( x n + b 1 ) in ⁢ B r / 20 ⁢ ( x 0 )

with

a 0 ≤ a 1 ≤ b 1 ≤ b 0 𝑎𝑛𝑑 b 1 - a 1 ≤ ( 1 - c ) ⁢ ε ⁢ r

and 0<c<1 universal.

If u satisfies (3.11) with, say r=1, then we can apply the Harnackinequality repeatedly and obtain

U β ⁢ ( x n + a m ) ≤ u ⁢ ( x ) ≤ U β ⁢ ( x n + b m ) in ⁢ B 20 - m ⁢ ( x 0 ) ,

with

b m - a m ≤ ( 1 - c ) m ⁢ ε

for all m’s such that

( 1 - c ) m ⁢ 20 m ⁢ ε ≤ ε ¯ .

This implies that for all such m’s, the oscillation of the renormalizedfunctions u~k in Br⁢(x0),r=20-m, is less than (1-c)m=20-γ⁢m=rγ. Thus, the following corollary holds.

Corollary 3.7

Let u~k be as defined in formula (3.9). Then

| u ~ k ⁢ ( x ) - u ~ k ⁢ ( x 0 ) | ≤ C ⁢ | x - x 0 | γ

for all x∈B1⁢(x0) such that |x-x0|≥εk/ε~.

Note now that

- 1 ≤ u ~ k ⁢ ( x ) ≤ 1 for ⁢ x ∈ B 1

and F⁢(uk) converges to B1∩{xn=0} in the Hausdorffdistance. These facts together with Ascoli–Arzelà theorem give that as εk→0 the graphs of the u~k converge(up to a subsequence) in the Hausdorff distance to the graph of a Höldercontinuous function u~ over B1/2.

Thus the improvement of flatness proof in the non-degenerate case can beconcluded.

3.4 Improvement of Flatness. Degenerate Case

In this case, the negative part of u is negligible and the positive partis close to a one-plane solution, i.e. for some ε>0 small, wehave

U 0 ⁢ ( x n - ε ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε ) in ⁢ B 1 .

This time the key lemma is the following.

Lemma 3.8

Assume that∥f∥L∞⁢(B1)≤ε4 and

(3.12) ∥ u - ∥ L ∞ ⁢ ( B 1 ) ≤ ε 2 .

There exists a universal r1 such that if 0<r≤r1 and 0<ε≤ε1 for someε1 depending on r, then

U 0 ⁢ ( x ⋅ ν 1 - r ⁢ ε 2 ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x ⋅ ν 1 + r ⁢ ε 2 ) in ⁢ B r ,

with |ν1|=1 and |ν1-en|≤C⁢ε for a universal constant C.

The proof follows the pattern of the non-degenerate case.

Fix r≤r1, to be chosen suitably. By contradiction assume that, forsome sequences εk→0 and uk, solutions of ourfree boundary problem in B1 with right-hand side fk, we have ∥fk∥L∞⁢(B1)≤εk4 and

∥ u k - ∥ L ∞ ⁢ ( B 1 ) ≤ ε k 2 ,

U 0 ⁢ ( x n - ε k ) ≤ u k ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε k ) in ⁢ B 1 , 0 ∈ F ⁢ ( u k ) ,

but the conclusion of the lemma does not hold.

Then one proves, via a one-phase version of the Harnack inequality inTheorem 3.6, that the sequence of normalized functions

u ~ k ⁢ ( x ) = u k ⁢ ( x ) - x n ε k , x ∈ B 1 + ⁢ ( u k ) ∪ F ⁢ ( u k )

converges to a limit function u~, Hölder continuous in B1/2. The limit function u~ is a viscosity solution of the linearizedproblem

{ Δ ⁢ u ~ = 0 in B 1 / 2 ∩ { x n > 0 } , u ~ x n = 0 on B 1 / 2 ∩ { x n = 0 } .

The regularity of u~ is not a problem and the contradictionargument proceeds as before with obvious changes.

Lemma 3.8 provides the first step in the flatness improvement. Noticethat this improvement is obtained through the closeness of the positivephase to a one-plane solution, as long as inequality (3.12)holds. This inequality expresses in another quantitative way the degeneracyof the negative phase and should be kept valid at each step of the iterationof Lemma 3.8. However, it could happen that this is not the case andin some step of the iteration, say at the level εk offlatness, the norm ∥u-∥L∞⁢(B1) becomes of orderεk2. When this occurs, a suitable rescaling restores anon-degenerate situation. This gives rise in the final iteration to a newdichotomy. The situation is precisely described in the following lemma.

Lemma 3.9

Let u be a solution in B1 satisfying

U 0 ⁢ ( x n - ε ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε ) in ⁢ B 1

with

∥ f ∥ L ∞ ⁢ ( B 1 ) ≤ ε 4 ,

and for C~ universal,

∥ u - ∥ L ∞ ⁢ ( B 2 ) ≤ C ~ ⁢ ε 2 , ∥ u - ∥ L ∞ ⁢ ( B 1 ) > ε 2 .

There exists (universal) ε1 such that, if0<ε≤ε1, the rescaling

u ε ⁢ ( x ) = ε - 1 / 2 ⁢ u ⁢ ( ε 1 / 2 ⁢ x )

satisfies, in B2/3,

U β ′ ⁢ ( x n - C ′ ⁢ ε 1 / 2 ) ≤ u ε ⁢ ( x ) ≤ U β ′ ⁢ ( x n + C ′ ⁢ ε 1 / 2 )

with β′∼ε2 and C′ depending on C~.

Let us see how the dichotomy arises. To prove our theorem in the degeneratecase, choose r¯ small (e.g., ≤116) and assume β<ε. From

U 0 ⁢ ( x n - ε ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε ) in ⁢ B 1 ,

since

∥ u - U β ∥ L ∞ ⁢ ( B 1 ) ≤ η ¯ = ε 3 ,

we infer

∥ u - ∥ L ∞ ⁢ ( B 1 ) ≤ β + ε 3 ≤ 2 ⁢ ε .

Set ε′=2⁢ε. Then

U 0 ⁢ ( x n - ε ′ ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x n + ε ′ ) in ⁢ B 1

and

∥ f ∥ L ∞ ⁢ ( B 1 ) ≤ ( ε ′ ) 4 , ∥ u - ∥ L ∞ ⁢ ( B 1 ) ≤ ( ε ′ ) 2 .

From Lemma 3.8, we get

U 0 ⁢ ( x ⋅ ν 1 - r ¯ ⁢ ε ′ 2 ) ≤ u + ⁢ ( x ) ≤ U 0 ⁢ ( x ⋅ ν 1 + r ¯ ⁢ ε ′ 2 ) in ⁢ B r ¯

with |ν1|=1 and |ν1-en|≤C⁢ε′ for a universal constant C.

We now rescale considering the blow-up sequence for k=1,2,…,

u k ⁢ ( x ) = u ⁢ ( r ¯ k ⁢ x ) r ¯ k , x ∈ B 1

and set εk=2-k⁢ε′,

f k ⁢ ( x ) = r ¯ k ⁢ f ⁢ ( r ¯ k ⁢ x ) , x ∈ B 1 .

Note that

∥ f k ∥ L ∞ ⁢ ( B 1 ) ≤ r ¯ k ⁢ ( ε ′ ) 4 ≤ 1 16 ⁢ ( ε ′ ) 4 = ε k 4 .

We can iterate Lemma 3.8 and obtain

U 0 ⁢ ( x ⋅ ν k - ε k ) ≤ u k + ⁢ ( x ) ≤ U 0 ⁢ ( x ⋅ ν k + ε k ) in ⁢ B 1

with |νk-νk-1|≤C⁢εk-1, as long as

∥ u k - ∥ L ∞ ⁢ ( B 1 ) ≤ ε k 2 .

Let k∗>1 be the first integer for which this fails:

∥ u k ∗ - ∥ L ∞ ⁢ ( B 1 ) > ε k ∗ 2

and

∥ u k ∗ - 1 - ∥ L ∞ ⁢ ( B 1 ) ≤ ε . k ∗ - 1 2

We also have

U 0 ⁢ ( x ⋅ ν k ∗ - 1 - ε k ∗ - 1 ) ≤ u k ∗ - 1 + ⁢ ( x ) ≤ U 0 ⁢ ( x ⋅ ν k ∗ - 1 + ε k ∗ - 1 ) in ⁢ B 1 .

By a usual comparison argument we can write

u k ∗ - 1 + ⁢ ( x ) ≤ C ⁢ | x n - ε k ∗ - 1 | ⁢ ε k ∗ - 1 2 in ⁢ B 19 / 20

for C universal. Rescaling, we have

∥ u k ∗ - ∥ L ∞ ⁢ ( B 1 ) ≤ C 1 ⁢ ε k ∗ 2

where C1 is universal (C1 depends on r¯). Then uk∗satisfies the assumptions of Lemma 3.9 and therefore the rescaling

v ⁢ ( x ) = ε k ∗ - 1 / 2 ⁢ u k ∗ ⁢ ( ε k ∗ 1 / 2 ⁢ x )

satisfies, in B2/3,

U β ′ ⁢ ( x ⋅ ν k ∗ - C ′ ⁢ ε k ∗ 1 / 2 ) ≤ v ⁢ ( x ) ≤ U β ′ ⁢ ( x ⋅ ν k ∗ + C ′ ⁢ ε k ∗ 1 / 2 )

with β′∼εk∗2. Set ε^=C′⁢εk∗1/2. Then v is asolution of our free boundary problem in B2/3 with right-hand side

g ⁢ ( x ) = ε k ∗ 1 / 2 ⁢ f k ∗ ⁢ ( ε k ∗ 1 / 2 ⁢ x )

and the flatness assumption

U β ′ ⁢ ( x ⋅ ν k ∗ - ε ^ ) ≤ v ⁢ ( x ) ≤ U β ′ ⁢ ( x ⋅ ν k ∗ + ε ^ ) .

Since β′∼εk∗2, we have

∥ g ∥ L ∞ ⁢ ( B 1 ) ≤ ε k ∗ 1 / 2 ⁢ ε k ∗ 4 ≤ ε ^ 2 ⁢ β ′

as long as ε is small enough. Under these restrictions, vsatisfies the assumptions of the non-degenerate case and we can proceedaccordingly.

4 Higher Regularity

4.1 Smoothness of Flat Free Boundaries

Assume now that f±∈C∞⁢(B1) or (real) analytic and,still, that the free boundary is flat. Thanks to the results of Section 3 we know that F⁢(u) is C1,γ. As a consequence u is aclassical solution, i.e. the free boundary condition is satisfied in a point-wise sense. The question is if also u and F⁢(u) are C∞ or (real) analytic, respectively. It is well known that if uis at least C2 up to the free boundary from both sides, through a zero-orderhodograph transformation and a suitable reflection map, as in [23, Theorem 3.2]it follows that u and F⁢(u) are C∞ or analytic. Thus the main point is to show that flat free boundaries areat least C2. Our main theorem gives indeed C2,γ∗regularity of flat free boundaries for a universal γ∗≤γ, provided f±∈C0,γ⁢(B1). Precisely:

Theorem 4.1

Let u be a (Lipschitz) viscosity solution to (1.1)in B1. There exists a universal constant η¯>0 such that, if

{ x n ≤ - η } ⊂ B 1 ∩ { u + ( x ) = 0 } ⊂ { x n ≤ η } for 0 ≤ η ≤ η ¯ ,

then F⁢(u) is C2,γ∗ in B1/2 for a small γ∗ universal, with the C2,γ∗ norm bounded by auniversal constant.

Having proved C2,γ∗ regularity, we can also proveintermediate Schauder estimates:

Theorem 4.2

Let k be a nonnegative integer. Assume that f±∈Ck,γ⁢(B1). Then F⁢(u)∩B1/2 is Ck+2,γ∗. If f± are C∞ or real analyticin B1, then F⁢(u)∩B1/2 is C∞ or realanalytic, respectively.

Indeed, Theorem 4.2 follows by a direct application of[27, Theorem 6.8.2], after transforming problem (1.1) into anelliptic system with coercive boundary conditions. This can be done as in[23]. We recall the main computations. For σ small, thepartial hodograph map

y ′ = x ′ , y n = u + ⁢ ( x )

is 1-1 from B1+¯⁢(u)∩Bσ⁢(0) ontoa neighborhood of the origin U⊂{yn≥0}, and flattens F⁢(u)into a set Σ⊂{yn=0}. The inverse mapping is the partialLegendre transformation

x ′ = y ′ , x n = ψ ⁢ ( y ) ,

where ψ satisfies yn=u+⁢(y′,ψ⁢(y)), y∈U. The free boundary is the graph of xn=ψ⁢(y′,0). Differentiating, we get

d ⁢ y n = ( ∇ ′ ⁡ u + + ∂ x n ⁡ u + ⁢ ∇ ′ ⁡ ψ ) ⋅ d ⁢ y ′ + ∂ x n ⁡ u + ⁢ ∂ y n ⁡ ψ ⁢ d ⁢ y n

from which

∂ x n ⁡ u + ⁢ ( y , ψ ⁢ ( y ) ) = 1 ∂ y n ⁡ ψ ⁢ ( y ) , ∇ ′ ⁡ u + ⁢ ( y , ψ ⁢ ( y ) ) = - ∇ ′ ⁡ ψ ⁢ ( y ) ∂ y n ⁡ ψ ⁢ ( y )

in U. Moreover, Δ⁢u+=f+ transforms into

ℱ 1 ( ψ ) : ≡ - ∂ y n ⁢ y n ⁡ ψ ( ∂ y n ⁡ ψ ) 3 + ∑ j = 1 n - 1 ( - ∂ y j ∂ y j ⁡ ψ ∂ y n ⁡ ψ + ∂ y j ⁡ ψ ∂ y n ⁡ ψ ∂ y n ∂ y j ⁡ ψ ∂ y n ⁡ ψ ) = f + ( y ′ , ψ ( y ) )

in U.

Concerning the negative part, let C be a constant larger than∂yn⁡ψ in U. Introduce the reflection map

x ′ = y ′ , x n = ψ ⁢ ( y ) - C ⁢ y n ,

which is 1-1 from a neighborhood of the origin U1⊆U onto B1-¯⁢(u)∩Bσ⁢(0) (choosing σ smaller, if necessary). Define, in U1,

φ ⁢ ( y ) = u - ⁢ ( y ′ , ψ ⁢ ( y ) - C ⁢ y n ) .

Differentiating, we get

∇ ′ ⁡ φ ⋅ d ⁢ y ′ + ∂ y n ⁡ φ ⁢ d ⁢ y n = ( ∇ ′ ⁡ u - + ∂ x n ⁡ u - ⁢ ∇ ′ ⁡ ψ ) ⋅ d ⁢ y ′ + ∂ x n ⁡ u - ⁢ ( ∂ y n ⁡ ψ - C ) ⁢ d ⁢ y n

from which

∂ x n ⁡ u - = ∂ y n ⁡ φ ∂ y n ⁡ ψ - C , ∇ ′ ⁡ u - = ∇ ′ ⁡ φ - ∂ y n ⁡ φ ∂ y n ⁡ ψ - C ⁢ ∇ ′ ⁡ ψ .

The equation Δ⁢u-=f- in B1-¯⁢(u)∩Bσ⁢(0) transforms into the equation

ℱ 2 ⁢ ( φ , ψ ) ≡ 1 ∂ y n ⁡ ψ - C ⁢ ∂ y n ⁡ ( ∂ y n ⁡ φ ∂ y n ⁡ ψ - C ) + ∑ j = 1 n - 1 ∂ y j ⁡ ( ∂ y j ⁡ φ - ∂ y n ⁡ φ ∂ y n ⁡ ψ - C ⁢ ∂ y j ⁡ ψ )

- ∑ j = 1 n - 1 ∂ y j ⁡ ψ ∂ y n ⁡ ψ - C ⁢ ∂ y n ⁡ ( ∂ y j ⁡ φ - ∂ y n ⁡ φ ∂ y n ⁡ ψ - C ⁢ ∂ y j ⁡ ψ ) = f - ⁢ ( y ′ , ψ ⁢ ( y ) - C ⁢ y n )

in U1.

Thus, in U1 we have the following nonlinear system:

{ ℱ 1 ⁢ ( ψ ) = f + ⁢ ( y ′ , ψ ⁢ ( y ) ) , ℱ 2 ⁢ ( φ , ψ ) = f - ⁢ ( y ′ , ψ ⁢ ( y ) - C ⁢ y n ) .

The free boundary conditions

u + = u - and | ∇ ⁡ u + | 2 - | ∇ ⁡ u - | 2 = 1 on ⁢ F ⁢ ( u )

become

{ φ ⁢ ( y ′ , 0 ) = 0 , 1 + | ∇ ′ ⁡ ψ ⁢ ( y ′ , 0 ) | 2 ( ∂ y n ⁡ ψ ⁢ ( y ′ , 0 ) ) 2 - ( ∂ y n ⁡ φ ⁢ ( y ′ , 0 ) ) 2 ( ∂ y n ⁡ ψ ⁢ ( y ′ , 0 ) - C ) 2 - ∥ ∇ ′ ⁡ φ ⁢ ( y ′ , 0 ) - ∂ n ⁡ φ ⁢ ( y ′ , 0 ) ∂ y n - C ⁢ ∇ ′ ⁡ ψ ⁢ ( y ′ , 0 ) ∥ ℝ n - 1 2 = 1 .

That is, after a simple computation,

{ φ ⁢ ( y ′ , 0 ) = 0 , ( 1 + | ∇ ′ ⁡ ψ ⁢ ( y ′ , 0 ) | 2 ) ⁢ ( 1 ( ∂ y n ⁡ ψ ⁢ ( y ′ , 0 ) ) 2 - ( ∂ y n ⁡ φ ⁢ ( y ′ , 0 ) ) 2 ( ∂ y n ⁡ ψ ⁢ ( y ′ , 0 ) - C ) 2 ) = 1 .

Linearization at y=0 gives (setting A=C-∂yn⁡ψ⁢(0))

{ ℒ 1 ⁢ ( ψ ) = | ∇ ⁡ u + ⁢ ( 0 ) | 2 ⁢ ∂ y n ⁢ y n ⁡ ψ + ∑ k = 1 n - 1 ∂ y k ⁢ y k ⁡ ψ = 0 , ℒ 2 ⁢ ( ψ , φ ) = 1 A 2 ⁢ ∂ y n ⁢ y n ⁡ φ + ∑ k = 1 n - 1 ∂ y k ⁢ y k ⁡ φ - | ∇ ⁡ u - ⁢ ( 0 ) | ⁢ ( 1 A 2 ⁢ ∂ y n ⁢ y n ⁡ ψ + ∑ k = 1 n - 1 ∂ y k ⁢ y k ⁡ ψ ) = 0 , ℬ 1 ⁢ ( φ ) = φ = 0 , ℬ 2 ⁢ ( ψ , φ ) = ( | ∇ ⁡ u + ⁢ ( 0 ) | 3 + 1 A ⁢ | ∇ ⁡ u - ⁢ ( 0 ) | 2 ) ⁢ ∂ y n ⁡ ψ - 1 A ⁢ | ∇ ⁡ u - ⁢ ( 0 ) | ⁢ ∂ y n ⁡ φ = 0 .

This system is elliptic with coercive boundary conditions under the naturalchoices of weights s1=s2=0, t1=t2=2for ℒ1 and ℒ2, respectively, and r1=-2, r2=-1 for ℬ1 and ℬ2, respectively. Indeed

order ℒ j = s j + t j = 2 ( j = 1 , 2 )

and

order ⁡ ℬ 1 = t 1 + r 1 = 0 , order ⁡ ℬ 2 = t 2 + r 2 = 1 .

4.2 From C1,γ toC2,γ∗. Outline and Strategy

The overall strategy for the proof of Theorem 4.1 is based againon a compactness argument leading to a limiting linearized problem in whichthe information for an improvement of flatness is stored. However, to reachthe C2,γ regularity requires a much more involved process becauseof the possible degeneracy of the negative part. Indeed this causes adelicate interplay between the two phases, as we shall try to explain below.We give here an idea of the complexity of the proof by outlining the overallstrategy. Ultimately the main source of difficulties is due to the presenceof a forcing term of general sign in the negative phase. Indeed, if f-≥0,the Hopf maximum principle would imply non-degeneracy (also) on thenegative side, making the two phases of comparable size and considerablysimplifying the final iteration procedure. It is worth noting that, even inthis easier scenario (and in particular in the homogeneous case), if onewants to attain uniform estimates with universal constants, then one mustemploy the more involved methods developed in [12] for the degeneratecase.

The first thing to do is to reinforce the notion of flatness, tailoring itfor the attainment of C2,γ regularity. This can be done byintroducing a suitable class of functions that we call two-phase andone-phase polynomials. In principle second-order polynomials shouldbe enough but it turns out that we need a small third-order perturbation.

Let ω∈ℝn, with |ω|=1, and let Sωbe an orthonormal basis containing ω. Let M∈Sn×nsatisfy M⁢ω=0. Define

P M , ω ⁢ ( x ) = x ⋅ ω - 1 2 ⁢ x T ⁢ M ⁢ x .

Set

V M , ω , a , b α , β ⁢ ( x ) = α ⁢ ( 1 + a ⋅ x ) ⁢ P M , ω + ⁢ ( x ) - β ⁢ ( 1 + b ⋅ x ) ⁢ P M , ω - ⁢ ( x ) , α > 0 , β ≥ 0 , a , b ∈ ℝ n .

These are our two-phase polynomials, one-phase if β=0. In theparticular case when M=0, a=b=0 and ω=en, we obtain the two-planefunction

U β ⁢ ( x ) = α ⁢ x n + - β ⁢ x n - .

The unit vector ω establishes the “direction offlatness”.

We shall need to work with a subclass, strictly related to problem (1.1), at least at the origin. We denote by 𝒱f± the classof functions of the form VM,ω,a,bα,β for which

2 ⁢ α ⁢ a ⋅ ω - α ⁢ t ⁢ r ⁢ M = f + ⁢ ( 0 )

2 ⁢ β ⁢ b ⋅ ω - β ⁢ t ⁢ r ⁢ M = f - ⁢ ( 0 ) if ⁢ β ≠ 0 ,

α 2 - β 2 = 1 , if ⁢ β ≠ 0 ,

and

α 2 ⁢ a ⋅ ω ⊥ = β 2 ⁢ b ⋅ ω ⊥ for all ⁢ ω ⊥ ∈ S ω .

The role of the last condition is to make VM,ω,a,bα,βan “almost” viscosity subsolution.

When β=0, then there is no dependence on b and a⋅ω⊥=0. Thus, we drop the dependence on β,b and f- in ournotation above and we indicate the dependence on aω:=a⋅ω.

We introduce the following definitions.

Definition 4.3

Let V=VM,ω,a,bα,β. We say that u is (V,ε,δ) flat in B1 if

V ⁢ ( x - ε ⁢ ω ) ≤ u ⁢ ( x ) ≤ V ⁢ ( x + ε ⁢ ω ) in ⁢ B 1

and

| a | , | b ′ | , ∥ M ∥ ≤ δ ⁢ ε 1 / 2 , | b n | ≤ δ 2 , | b n | ⁢ ∥ M ∥ ≤ δ 2 ⁢ ε .

Given V=VM,ω,a,bα,β, set

V r ⁢ ( x ) = V ⁢ ( r ⁢ x ) r

and notice that

V r = V r ⁢ M , ω , r ⁢ a , r ⁢ b α , β .

Definition 4.4

Let V=VM,ω,a,bα,β. We say that u is (V,ε,δ) flat in Br if the rescaling

u r ⁢ ( x ) := u ⁢ ( r ⁢ x ) r

is (Vr,εr,δ) flat in B1.

Notice that if u is (V,ε,δ) flat in Br then

V ⁢ ( x - ε ⁢ ω ) ≤ u ⁢ ( x ) ≤ V ⁢ ( x + ε ⁢ ω ) in ⁢ B r .

The parameter ε measures the level of polynomial approximation andδ is a flatness parameter (also controlling the C0,γnorms of f+ and f-).

To obtain uniform point-wise C2,γ∗ regularity both for thesolution and the free boundary in B1/2 we have to show that u is (Vk,λk2+γ∗,δ) flat in Bλkfor λk=ηk and all k≥0, for some δ,ηsmall and a sequence of Vk converging to a final profile V0.

The starting point in the proof of Theorem 4.1 is to show thatthe flatness condition (3.1) allows us to normalize our solution sothat a rescaling ur¯ of u is close to a one- or two-phasepolynomial. This kind of dichotomy parallels in a sense what happens in theflatness toC1,γ case but at a quadratic order of approximation. Set

u r ⁢ ( x ) := u ⁢ ( r ⁢ x ) r , f ± r ⁢ ( x ) = r ⁢ f ± ⁢ ( r ⁢ x ) , x ∈ B 1 .

Lemma 4.5

There exist universal constants ε¯,δ¯,λ¯such that if u satisfies (3.1) with η¯=η¯⁢(ε¯) then either of these flatness conditions holds with r¯=r¯⁢(ε¯).

Non-degenerate case: u r ¯ is ( V , λ ¯ 2 + γ , δ ¯ ) flat in B 1 , with V = V 0 , e n , a , b α , β ∈ 𝒱 f ± ,

a ′ = b ′ = 0 , β ≥ 1 2 ⁢ δ ¯ 1 / 2 ⁢ λ ¯ 2 + γ ,

and

| f + r ¯ ⁢ ( x ) - f + r ¯ ⁢ ( 0 ) | ≤ δ ¯ ⁢ | x | γ , | f - r ¯ ⁢ ( x ) - f - r ¯ ⁢ ( 0 ) | ≤ β ⁢ δ ¯ ⁢ | x | γ .
Degenerate case: u r ¯ + is ( V , λ ¯ 2 + γ , δ ¯ ) flat in B 1 , for V = V 0 , e n , a n 1 ∈ 𝒱 f + ,

| u r ¯ - + 1 2 ⁢ f - r ¯ ⁢ ( 0 ) ⁢ x n 2 | ≤ δ ¯ 1 / 2 ⁢ λ ¯ 2 + γ in ⁢ B 1 - ⁢ ( u r ¯ )

and

∥ f - r ¯ ∥ ∞ ≤ δ ¯ , | f ± r ¯ ⁢ ( x ) - f ± r ¯ ⁢ ( 0 ) | ≤ δ ¯ ⁢ | x | γ .

We describe the dichotomy as follows.

Case 1. It corresponds to a non-degenerate configuration, in which the twophases have comparable size and ur¯ is trapped between twotranslations of a genuine two-phase polynomials, with a positiveslope β (not too small).

Case 2. It corresponds to a degenerate configuration, where the negativephase that has either zero slope or a small one (but not negligible) withrespect to ur¯+ and ur¯+ is trapped between twotranslations of a one-phase polynomial. Note that this situation cannotoccur if f-≥0 unless u- is identically zero.

Next we examine how the initial flatness corresponding to cases 1 and 2above improves successively at a smaller scale. We construct the followingtwo “subroutines”, to be implemented inthe course of the final iteration towards C2,γ∗ regularity.

The first subroutine provides a two-phaseC2,γ flatnessimprovement: if u is (V,λ¯2+γ,δ¯) flat in Bλ then u is (V¯,(η⁢λ)2+γ,δ¯) flat in Bλ⁢η, with V¯ close to V.This result applies to the non-degenerate case.

Proposition 4.6

Proposition 4.6 (Two-Phase Flatness Improvement)

There exist η¯,δ¯,λ¯ universal such thatif for β>0,u is (V,λ2+γ,δ¯) flat in Bλ, λ≤λ¯with V=VM,en,a,bα,β∈Vf±,

| f + ⁢ ( x ) - f + ⁢ ( 0 ) | ≤ δ ¯ ⁢ | x | γ , | f - ⁢ ( x ) - f - ⁢ ( 0 ) | ≤ β ⁢ δ ¯ ⁢ | x | γ

and

| ∇ ⁡ u + | 2 - | ∇ ⁡ u - | 2 = 1 on ⁢ F ⁢ ( u ) ∩ B 2 / 3 ⁢ λ ,

then u is (V¯,(η¯⁢λ)2+γ,δ¯) in Bη¯⁢λwithV¯=VM¯,ν¯,a¯,b¯α¯,β¯∈Vf±and|β-β¯|≤C⁢λ1+γ for C universal.

The second subroutine provides a one-phase flatness improvementthat will be used when we will deal with the degenerate case, that is whenthe flatness of the free boundary only guarantees closeness of the positivepart u+ to a quadratic profile. More precisely if u+ is (V,λ¯2+γ,δ¯) flat in Bλ and |∇⁡u+| is close to α on F⁢(u), thenu+ enjoys a C2,γ flatness improvement, with V¯ closeto V.

Proposition 4.7

Proposition 4.7 (One-Phase Flatness Improvement)

There exist η¯,δ¯,λ¯such that if for β=0, u+ is (V,λ2+γ,δ¯)flat in Bλ,λ≤λ¯ withV=VM,en,anα∈Vf+,

| f + ⁢ ( x ) - f + ⁢ ( 0 ) | ≤ δ ¯ ⁢ | x | γ

and

| | ∇ ⁡ u + | - α | ≤ δ ¯ 1 / 2 ⁢ λ 1 + γ on ⁢ F ⁢ ( u ) ∩ B 2 / 3 ⁢ λ ,

in the viscosity sense, then u+ is (V¯,(η¯⁢λ)2+γ,δ¯)in Bη¯⁢λ withV¯=VM¯,ν¯,a¯ν¯α∈Vf+.

The achievement of the improvements above relies on a higher orderrefinement of the Harnack inequalities in Section 3. This gives thenecessary compactness to pass to the limit in a sequence of renormalizationsof u of the type (e.g., in the genuine two-phase case)

v ~ ε ⁢ ( x ) = { v ⁢ ( x ) - α ⁢ ( 1 + a ⋅ x ) ⁢ P M , e n α ⁢ ε , x ∈ B 1 + ⁢ ( u ) ∪ F ⁢ ( u ) , v ⁢ ( x ) - β ⁢ ( 1 + b ⋅ x ) ⁢ P M , e n β ⁢ ε , x ∈ B 1 - ⁢ ( u ) , β > 0 , 0 , x ∈ B 1 - ⁢ ( u ) , β = 0 ,

and to obtain a limiting transmission or Neumann problem, which turns out to bethe same as in Section 3. From the regularity of the solution ofthis problem we get the information to improve the two-phase or one-phaseapproximation for u or u+, respectively, and hence their flatness.

4.3 The New Dichotomy

Now we can start iterating. As we have seen, according to Case 1 above,after a suitable rescaling, we face a first dichotomy “degenerate versus non-degenerate”.

In the latter case the two-phase subroutine of Proposition 4.6 can beapplied indefinitely to reach pointwise C2,γ∗ regularityfor some universal γ∗.

When u falls into the degenerate case, a new kind of dichotomyappears. First of all, to run the one-phase subroutine inProposition 4.7 we need to make sure that the closeness of u- toa purely quadratic profile makes u+ to be a (viscosity) solution of aone-phase free boundary problem with |∇⁡uν+| close to an appropriate α on F⁢(u).At this point two alternatives occur at a smaller scale:

either u- is closer to a purely quadratic profile at aproper C2,γ rate and u+ enjoys a C2,γ flatnessimprovement;
or u- is closer (at a C2,γ rate) to a one-phasepolynomial profile with a small non-zero slope but u+ only enjoys an“intermediate” C2 flatnessimprovement.

To give a precise statement it is convenient to introduce a new class 𝒬f- of functions, defined as

Q p , q , ω , M = ( x ⋅ ω - 1 2 ⁢ x T ⁢ M ⁢ x ) ⁢ ( p + q ⋅ x ) - 1 2 ⁢ ( f - ⁢ ( 0 ) + p ⁢ tr ⁡ M ) ⁢ ( x ⋅ ω ) 2 ,

with p∈ℝ,q∈ℝn,M∈Sn×n, such that

q ⋅ ω = 0 , M ⁢ ω = 0 , ∥ M ∥ ≤ 1 .

In the degenerate case, we use these functions to approximate u- in a C2,γ fashion at a smaller and smaller scale. We have:

Proposition 4.8

There exist universal constants λ¯,δ¯,η¯ suchthat if u+ is (V,λ2+γ,δ¯) flat inBλ,λ≤λ¯ withV=VM,en,an1∈Vf+,

| f ± ⁢ ( x ) - f ± ⁢ ( 0 ) | ≤ δ ¯ ⁢ | x | γ , ∥ f - ∥ ∞ ≤ δ ¯

and

| u - - Q 0 , 0 , e n , 0 | ≤ δ ¯ 1 / 2 ⁢ λ 2 + γ in ⁢ B λ - ⁢ ( u ) ,

then either one of the following holds:

There exists V ¯ = V M ¯ , 𝐞 ¯ , a ¯ 𝐞 ¯ 1 ∈ 𝒱 f + such that u + is ( V ¯ , ( η ¯ ⁢ λ ) 2 + γ , δ ¯ ) flat in B η ¯ ⁢ λ , and

| u - - Q 0 , 0 , 𝐞 ¯ , 0 | ≤ δ ¯ 1 / 2 ⁢ ( η ¯ ⁢ λ ) 2 + γ in ⁢ B η ¯ ⁢ λ - ⁢ ( u ) .
There exists V ∗ = V M ∗ , 𝐞 ∗ , a 𝐞 ∗ ∗ α ∗ ∈ 𝒱 f + such that u + is ( V ∗ , η ¯ 2 ⁢ λ 2 + γ , δ ¯ ) flat in B η ¯ ⁢ λ , and

| u - - Q p ∗ , q ∗ , 𝐞 ∗ , M ∗ | ≤ δ ¯ 1 / 2 ⁢ ( η ¯ ⁢ λ ) 2 + γ in ⁢ B η ¯ ⁢ λ - ⁢ ( u ) ,

for ( α ∗ ) 2 - ( p ∗ ) 2 = 1 and p ∗ < 0 , |p∗|∼(δ¯1/2⁢λ1+γ), |q∗|=O⁢(δ¯1/2⁢λγ).

If (D1) occurs indefinitely, we are done. If it does not, we prove that theintermediate improvement in (D2) is kept for a while, at smaller and smallerscale. The final and crucial step is to prove that, at a given universallysmall enough scale, the C2,γ one-phase approximation of u-,together with the intermediate C2 flatness improvement of u+, isgood enough to recover a full C2,γ∗ two-phase improvementof u with a universal γ∗<γ.

More precisely, at the beginning u+ is (V,λ¯2+γ,δ¯) flat while u- is C2,γ close to a purequadratic profile. This closeness improves at a C2,γ rate until(possibly) the slope of the approximating polynomial Q is no longer zero,say at scale λ. However, to obtain the desired full flatness of u, we need to reach a scale ρ=λ⁢r for r∼λ1+1/γ (see the proposition below). It is necessary to exploit also theinformation that the flatness of u+ is in fact improving at a C2rate for a little while, hence allowing us to continue the iteration on thenegative side and to obtain that u- is C2,γ close to anon-degenerate configuration at an even smaller scale. We have seen that inthe case of the C1,γ estimates this issue is not present. Thekey result is the following:

Proposition 4.9

There exist λ¯,δ¯,γ∗ universal such thatif u+ is (V,r2⁢λ2+γ,δ¯) flat in Br⁢λ,λ≤λ¯ with V=VM,en,anα∈Vf+, for r suchthat δ¯1/2⁢rγ∈[2⁢η¯γ⁢λ1+γ,2⁢λ1+γ), and

| u - - Q p , q , e n , M | ≤ δ ¯ 1 / 2 ⁢ ( r ⁢ λ ) 2 + γ in ⁢ B r ⁢ λ - ⁢ ( u ) ,

for α2-p2=1 and p<0,|p|∼δ¯1/2⁢λ1+γ,|q|=O⁢(δ¯1/2⁢λγ), thenu is (V¯,(r⁢λ)2+γ∗,δ¯) flatin Br⁢λ withV¯=VM,en,a,bα,β∈Vf±, β=|p|.

From this point on we can go back to the two-phase subroutine to reachpointwise C2,γ∗ regularity.

5 Generalization and Further Developments

The results in Sections 3 and 4 extend without much effortto more general linear uniformly elliptic equations with C0,γ (C∞, analytic) coefficients and to more general free boundary jump conditions

| ∇ ⁡ u + | = G ⁢ ( | ∇ ⁡ u - | , ν , x ) ,

where G is C2 (C∞, analytic)with respect to all its arguments. For these operators, the theory ofviscosity solutions to inhomogeneous free boundary problems has reached aconsiderable level of completeness.

For problems governed by fully nonlinear operators, we proved in [10] that for a fairly general class of operators, Lipschitz viscositysolutions with Lipschitz or flat (in the sense of (3.1)) freeboundaries are indeed classical (C1,γ). The questions ofLipschitz continuity of solutions and higher regularity of the free boundaryremain open problems.

Strong regularity of the free boundary for homogeneous problems governed bythe p-Laplace operator has been developed by Lewis and Nystrom in [25, 26]. Nothing is known in presence of distributed sources.

Also of great importance, we believe, is to have information on theHausdorff measure or dimension of the singular (nonflat)points of the free boundary. For instance, in three and four dimensions, the freeboundary for local energy minimizer in the variational problem

∫ Ω { | ∇ ⁡ u | 2 + χ { u > 0 } } → min

is a smooth surface (see [7, 22]). In dimension n=7, De Silva andJerison in [14] provided an example of a minimizer with singular freeboundary. Thus the conjecture is that energy minimizing free boundariesshould be smooth for n<7.

Nothing is known in the nonhomogeneous case.

Dedicated to Ireneo for his 70th birthday with all our friendship

Communicated by Antonio Ambrosetti and David Arcoya

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Received: 2017-02-03

Revised: 2017-03-07

Accepted: 2017-03-09

Published Online: 2017-04-21

Published in Print: 2017-05-01

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2017-0015

Keywords for this article

Two-Phase Free Boundary Problem; Free Boundary Regularity; Perron Solutions, Elliptic Systems

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