Global solutions to nonlinear two-phase free boundary problems

TL;DR

This paper classifies all global Lipschitz solutions to certain two-phase free boundary problems governed by fully nonlinear equations, identifying them as either two-plane solutions or solutions to a one-phase problem.

Contribution

It provides a complete classification of global solutions for a class of nonlinear two-phase free boundary problems, extending understanding of their structure.

Findings

Global Lipschitz solutions are either two-plane solutions or one-phase solutions.

Classification applies to problems governed by concave fully nonlinear equations.

Results enhance understanding of free boundary problem solution structures.

Abstract

We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations, as either two-plane solutions or solutions to a one-phase problem.

Equations

$$J(u)={\int }_{\Omega }|\nabla u{|}^2+{\mathcal{H}}^n({\chi }_{\{u>0\}})dx,$$

$$\left\{\begin{array}[]{ll}\triangle u=0,&\hbox{in $\{u\neq 0\}$}\\ &\\ (u_{\nu}^{+})^{2}-(u_{\nu}^{-})^{2}=1,&\hbox{on $\Gamma(u):=\{u=0\}.$}\\ \end{array}\right.$$

$$\Phi (r):={\left(\frac{1}{r^2}{\int }_{B_r}\frac{|\nabla u^{+}{|}^2}{|x{|}^{n-2}}dx\right)}^{\frac{1}{2}}{\left(\frac{1}{r^2}{\int }_{B_r}\frac{|\nabla u^{-}{|}^2}{|x{|}^{n-2}}dx\right)}^{\frac{1}{2}}$$

$$\Phi (1)\ge \Phi (0+)\simeq u_{\nu }^{+}(0)\cdot u_{\nu }^{-}(0),$$

$$p_{a,b}:=a(x\cdot \nu {)}^{+}-b(x\cdot \nu {)}^{-},a^2-b^2=1,a,b>0,$$

$$u_{\nu }^{+}=G(u_{\nu }^{-}).$$

$$\left\{\begin{array}[]{ll}\mathcal{F}(D^{2}u)=0,&\hbox{in $B_{1}^{+}(u)\cup B_{1}^{-}(u),$}\\ &\\ u_{\nu}^{+}=G(u_{\nu}^{-}),&\hbox{on $\Gamma(u):=\partial B^{+}(u)\cap B_{1}.$}\\ \end{array}\right.$$

$$B_1^{+}(u):=\{x\in B_1:u(x)>0\},B_1^{-}(u):=\{x\in B_1:u(x)\le 0{\}}^{\circ },$$

$$\mathcal{F}$$

$$u=a((x-x_0)\cdot \nu {)}^{+}-b((x-x_0)\cdot \nu {)}^{-}\text{mbox}witha,b>0,a=G(b),$$

$$u^{-}\equiv 0,$$

$$a_{ij}\left(\frac{\nabla u}{|\nabla u|}\right)u_{ij}=0,$$

$$D^2{u}^{+}(0)=diag({\kappa }_1{u}_n^{+}(0),\dots ,{\kappa }_{n-1}{u}_n^{+}(0),u_{nn}^{+}(0))$$

$$D^2{u}^{-}(0)=diag(-{\kappa }_1{u}_n^{-}(0),\dots ,-{\kappa }_{n-1}{u}_n^{-}(0),u_{nn}^{-}(0)).$$

$$\mathcal{F}\left(diag\left({\kappa }_1,\dots ,{\kappa }_{n-1},\frac{u_{nn}^{+}(0)}{u_n^{+}(0)}\right)\right)=\mathcal{F}\left(diag\left({\kappa }_1,\dots ,{\kappa }_{n-1},-\frac{u_{nn}^{-}(0)}{u_n^{-}(0)}\right)\right)=0.$$

$$\frac{u_{nn}^{+}(0)}{u_n^{+}(0)}=-\frac{u_{nn}^{-}(0)}{u_n^{-}(0)}.$$

$$u_{nn}^{+}(0),u_{nn}^{-}(0)<0,$$

$$P_{M,\nu }(x):=x\cdot \nu +\frac{1}{2}{x}^{T}Mx,$$

$$\text{M∈Sn×n such that Mν=0 and F(M)=0}.$$

$$|u(x)-P_{M,e_n}(x)|\le ϵ\text{in B1},ϵ\le {ϵ}_0$$

$$\Vert M\Vert \le {\delta }_0{ϵ}^{1/2},$$

$$|u-P_{\bar{M},\nu }|\le ϵr_0^{2+{\alpha }_0},\text{in Br0}$$

$$\Vert M-\bar{M}\Vert \le Cϵ$$

$$|\nabla u^{+}(0)|\le 1-c_0ϵ.$$

$${\mathcal{M}}_{\lambda ,\Lambda }^{-}(N)\le \mathcal{F}(M+N)-\mathcal{F}(M)\le {\mathcal{M}}_{\lambda ,\Lambda }^{+}(N).$$

$${\mathcal{M}}_{\lambda ,\Lambda }^{-}(N)=\Lambda \sum _{{\mu }_i<0}{\mu }_i+\lambda \sum _{{\mu }_i>0}{\mu }_i,{\mathcal{M}}_{\lambda ,\Lambda }^{+}(N)=\lambda \sum _{{\mu }_i<0}{\mu }_i+\Lambda \sum _{{\mu }_i>0}{\mu }_i,$$

$$u(x)\ge \phi (x)(\text{resp. u(x)≤φ(x)})\text{in a neighborhood O of x0.}$$

$$\mathcal{F}(D^{2}v)>0(\text{resp}.<0)\text{in O,}$$

$$\mathcal{F}(D^{2}u)=0\text{in O,}$$

$$u_{\nu }^{+}=G(u_{\nu }^{-}),$$

Full text

O. S. is supported by NSF grant DMS-1200701.

1. Introduction

The classical two-phase free boundary problem studies critical functions $u$ of the energy

[TABLE]

which appears in various models of fluid mechanics or heat conduction (see for example [ACF, LW]).

The critical functions are harmonic except on their [math] level set where a jump condition on $\nabla u$ is imposed due to the presence of the discontinuous second term in the energy. Precisely, the Euler-Lagrange equation reads

[TABLE]

Here $u_{\nu}^{+}$ and $u_{\nu}^{-}$ denote the normal derivatives in the inward direction to the positivity sets of $u^{+}$ and $u^{-}$ respectively, and $\Gamma(u)$ is the free boundary of $u$.

The Lipschitz continuity of solutions to the free boundary problem (1.1) was obtained by Alt, Caffarelli and Friedman in [ACF]. They discovered a monotonicity formula relating two nonnegative harmonic functions $u^{\pm}$ defined in disjoint domains $\Omega^{\pm}$ in say $B_{2}$, that vanish on the boundary $\partial\Omega^{\pm}\cap B_{1}$. Precisely, the formula states that

[TABLE]

is monotone increasing in $r$, and $\Phi$ is constant if and only if $u^{+}$ and $u^{-}$ are linear functions. In the context of (1.1), this formula implies that the energy at scale 1 bounds the product of the slopes at a free boundary point $0\in\Gamma(u),$

[TABLE]

and then it follows from the free boundary condition that $u^{+}_{\nu}(0)$ is bounded above.

The Lipschitz continuity of solutions to (1.1) is a crucial ingredient also in the study of the regularity of $\Gamma(u)$. Indeed, after a blow-up analysis, the question of the regularity of $\Gamma(u)$ can be reduced to the classification of global Lipschitz solutions (see Caffarelli [C2]). Another consequence of the ACF monotonicity formula (by letting $r\to\infty$) is the classification of global “purely two-phase” solutions (i.e. with $u^{-}\not\equiv 0$) as the trivial two-plane solutions

[TABLE]

for some unit direction $\nu$. This implies that the only types of singularities for the free boundary $\Gamma(u)$ are the ones that occur in the one-phase setting when $u^{-}\equiv 0$.

The ACF monotonicity formula has been extensively used in various other contexts, however it is specific to the Laplace operator (see also [CJK, MP]). In [DS] and [CDS] we investigated the questions of Lipschitz regularity of solutions for two-phase free boundary problems governed by fully nonlinear operators $\mathcal{F}(D^{2}u)$ and with a general isotropic free boundary condition

[TABLE]

In [DS] we obtained the Lipschitz continuity of solutions under the assumption that $G(t)$ behaves like $t$ for all $t$ large, and this condition is satisfied for example in the two-phase problem (1.1). If in addition $\mathcal{F}$ is homogenous, it suffices to require $G(t)/t\to c_{0}$ as $t\to\infty$ for some constant $c_{0}$.

It turns out that in dimension $n=2$ the results can be improved significantly. This was shown by the authors in collaboration with Caffarelli in [CDS] where the Lipschitz continuity and the classification of global “purely two-phase” solutions were obtained for linear equations with measurable coefficients under very general free boundary conditions $u_{\nu}^{+}=G(u_{\nu}^{-},\nu,x)$.

In this paper we continue the study of the classification of global Lipschitz two-phase solutions in dimension $n\geq 3$. As mentioned above this problem is intimately connected to the regularity of the free boundary $\Gamma(u)$. In the nonlinear case the $C^{1,\alpha}$ regularity of $\Gamma(u)$ under perturbative assumptions was obtained by several authors, in slightly different settings (see for example [DFS, F1, Fe1, W1, W2].)

We consider the two-phase free boundary problem governed by a fully nonlinear uniformly elliptic operator $\mathcal{F}$

[TABLE]

Here $B_{r}\subset\mathbb{R}^{n}$ denotes the ball of radius $r$ centered at [math] and

[TABLE]

while as noted above, $u_{\nu}^{+}$ and $u_{\nu}^{-}$ denote the normal derivatives in the inward direction to $B_{1}^{+}(u)$ and $B_{1}^{-}(u)$ respectively. The function $G:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is of class $C^{1}$ and it satisfies the usual ellipticity assumption, that is $G(t)$ is strictly increasing.

Our main result is a Liouville theorem for global “two-phase” Lipschitz solutions.

Theorem 1.1**.**

Let $u$ be a Lipschitz continuous viscosity solution to (1.2) in $\mathbb{R}^{n}$. Assume that

[TABLE]

Then either $u$ is a two plane-solution

[TABLE]

or

[TABLE]

which means that $u$ solves the one-phase problem for $\mathcal{F}$.

Theorem 1.1 can be extended to more general operators $\mathcal{F}(D^{2}u,\nabla u,u)$ that depend also on $\nabla u$ and $u$ if appropriate assumptions are imposed on $\mathcal{F}$. Here we only state a version that applies to the $p$-Laplace equation. We consider quasilinear equations of the type

[TABLE]

with uniformly elliptic coefficients $a_{ij}\in C^{1}(S^{n-1})$. We remark that in [DS] we established also the Lipschitz continuity of solutions to the two-phase free boundary problem governed by (1.6) (see also the paper of Dipierro and Kharakhian [DK] for the case of minimizers in the $p$-Laplace equation).

Theorem 1.2**.**

The conclusion of Theorem 1.1 holds if $u$ is a global Lipschitz continuous viscosity solution to (1.2) governed by equation (1.6) instead of $\mathcal{F}$.

We roughly outline a formal idea of the proof of Theorem 1.1. Notice that $|\nabla u^{+}|$ is a subsolution of the linearized equation in $\{u>0\}$ and then its supremum occurs on the boundary $\Gamma(u)$. Assume for the moment that $\Gamma(u)$ is of class $C^{2}$ and that $|\nabla u^{+}|$ achieves its maximum at a point, say at $0\in\Gamma(u)$. Suppose the normal to $\Gamma(u)$ at 0 (pointing in the positive phase) is $e_{n}$ and let $\kappa_{1},\ldots,\kappa_{n-1}$ be the principal curvatures of $\Gamma(u)$ at [math]. Then the mixed derivatives vanish, i.e. $u^{+}_{in}(0)=0$, $i<n$,

[TABLE]

and similarly (in view of the free boundary condition)

[TABLE]

Using that $\mathcal{F}(D^{2}u^{+}(0))=\mathcal{F}(-D^{2}u^{-}(0))=0$ and that $\mathcal{F}$ is homogenous of degree one we have

[TABLE]

which by ellipticity of $\mathcal{F}$ gives

[TABLE]

On the other hand, by Hopf lemma applied to $u_{n}^{+}$, $u_{n}^{-}$ we have

[TABLE]

(unless $u_{n}^{+}$ and $u_{n}^{-}$ are constant) and we reach a contradiction.

The rigorous proof requires somewhat involved and technical arguments. One of the main steps consists in obtaining a weak Evans-Krylov type estimate for a nonlinear transmission problem. The assumptions that $\Gamma(u)$ is of class $C^{2}$ (even if $\mathcal{F}$ is concave) and that $|\nabla u^{+}|$ achieves its maximum at a point (rather than at infinity) cannot be justified. One major difficulty is that $\Gamma(u)$ is not known to be better than $C^{1,\alpha}$ even in the perturbative setting.

The idea of proof of Theorem 1.1 is to show a “reversed” improvement of flatness for the solution $u$, which means that if $u$ is sufficiently close to a two plane solution at a small scale then it remains close to the same two-plane solution at all larger scales. The key ingredient in the proof of our main Theorem 1.1 is the Proposition 1.3 below.

Since $\mathcal{F}$ is homogeneous of degree one, we can multiply $u^{+}$ and $u^{-}$ by suitable constants and assume that $G(1)=1$ and $\|\nabla u\|_{L^{\infty}}=1$. Denote by $P_{M,\nu}$ quadratic approximations of slope 1 to our free boundary problem (1.2),

[TABLE]

with $\nu$ a unit direction and

[TABLE]

Throughout the paper constants depending on $n,$ the ellipticity constants $\lambda,\Lambda$ of $\mathcal{F}$, and the modulus of continuity $\omega$ of $G^{\prime}$ on $[0,2]$, are called universal constants.

Proposition 1.3 is a dichotomy result for solutions $u$ which are $\epsilon$-perturbations of polynomials $P_{M,\nu}$ at scale 1. It says that either $u$ can be approximated by another polynomial $P_{\bar{M},\bar{\nu}}$ in a $C^{2,\alpha}$ fashion at a smaller scale, or that $|\nabla u(0)|$ has to be strictly below 1 an amount of order $\epsilon$.

Proposition 1.3** (Nonlinear Dichotomy).**

Assume that $0\in F(u)$, $G(1)=1$ and $|\nabla u|\leq 1$. There exist small universal constants, $\epsilon_{0},\delta_{0},r_{0},c_{0},\alpha_{0}>0$ such that if

[TABLE]

with

[TABLE]

then one of the following alternatives holds:

(i)

[TABLE]

for some $\bar{M},\nu$ with

[TABLE]

and $C$ universal;

(ii)

[TABLE]

We remark that assumption (1.8) implies that $\Gamma(u)\in C^{1,\alpha}$ and $u$ is a classical solution, hence $\nabla u^{+}(0)$ is well defined (see Section 2).

The paper is organized as follows. In Section 2 we provide notation and definitions, and we recall the flatness result established in [DFS]. The proof of the main Theorem 1.1 is presented in Section 3, assuming that Proposition 1.3 holds. In Section 4 we study the transmission problem which appears as the linearization to our free boundary problem, and we prove a linear version of Proposition 1.3. In the last section we provide the proof of Proposition 1.3 and we also sketch a proof of Theorem 1.2.

2. Preliminaries

In this section we present some preliminary definitions and known results.

First we define viscosity solutions to (1.2). Recall that $\mathcal{F}:\mathcal{S}^{n\times n}\to\mathbb{R}$ is uniformly elliptic if there exist $0<\lambda\leq\Lambda$ such that for every $M,N\in\mathcal{S}^{n\times n},$

[TABLE]

Here $\mathcal{S}^{n\times n}$ denotes the set of real $n\times n$ symmetric matrices and $\mathcal{M}^{\pm}_{\lambda,\Lambda}$ the extremal Pucci operators

[TABLE]

with $\mu_{i}$ denoting the eigenvalues of $N$.

The class of all uniformly elliptic operators with ellipticity constants $\lambda,\Lambda$ and such that $\mathcal{F}(0)=0$ will be denoted by $\mathcal{E}(\lambda,\Lambda).$

Given $u,\varphi\in C(B_{1})$, we say that $\varphi$ touches $u$ by below (resp. above) at $x_{0}\in B_{1}$ if $u(x_{0})=\varphi(x_{0}),$ and

[TABLE]

If this inequality is strict in $O\setminus\{x_{0}\}$, we say that $\varphi$ touches $u$ strictly by below (resp. above).

Let $\mathcal{F}\in\mathcal{E}(\lambda,\Lambda)$. If $v\in C^{2}(O)$, $O$ open subset in $\mathbb{R}^{n},$ satisfies

[TABLE]

we call $v$ a (strict) classical subsolution (resp. supersolution) to the equation $\mathcal{F}(D^{2}v)=0$ in $O$.

We recall that $u\in C(O)$ is a viscosity solution to

[TABLE]

if $u$ cannot be touched by below (resp. above) by a strict classical subsolution (resp. supersolution) at a point $x_{0}\in O.$ Similarly we can define viscosity subsolutions/supersolutions.

We refer the reader to [CC] for a comprehensive treatment of the theory of fully nonlinear elliptic equations. In particular, if $u$ is a viscosity solution to (2.1) then $u\in C^{1,\alpha}$ and any directional derivative $u_{e}$ belongs to the class ${\mathcal{S}}(\lambda,\Lambda)$ of “solutions” to linear equation with measurable coefficients, and therefore satisfies the Harnack inequality.

If in addition $\mathcal{F}$ is concave then $u\in C^{2,\alpha}$ by the Evans-Krylov theorem, and any second directional derivative $u_{ee}$ belongs to the class of subsolutions $\underline{\mathcal{S}}(\lambda,\Lambda)$ to linear equation with measurable coefficients, and therefore it satisfies the Weak Harnack inequality.

We now turn to the free boundary condition.

Definition 2.1**.**

We say that $u$ satisfies the free boundary condition

[TABLE]

at a point $y_{0}\in\Gamma(u)$ if for any unit vector $\nu$, there exists no function $\psi\in C^{2}$ defined in a neighborhood of $y_{0}$ with $\psi(y_{0})=0$, $\nabla\psi(y_{0})=\nu$ such that either of the following holds:

(1) $a\psi^{+}-b\psi^{-}\leq u$ with $a>0$, $b>0$ and $a>G(b)$ (i.e. $u$ is a supersolution);

(2) $a\psi^{+}-b\psi^{-}\geq u$ with $a>0$, $b>0$ and $a<G(b)$ (i.e. $u$ is a subsolution).

We only use comparison functions which cross the [math] level set transversally and therefore have a nontrivial negative part. For this reason the free boundary condition is preserved when taking uniform limits. It is straightforward to check that a uniform limit of solutions of (1.2) satisfies (1.2) as well (see Lemma 2.3 in [DS]).

Next we state the flatness theorem obtained by De Silva, Ferrari and Salsa in [DFS].

Definition 2.2**.**

A two-plane solution $U$ to (1.2) is given by

[TABLE]

for some $\nu\in S^{1}$ and with

[TABLE]

The following result was established in [DFS].

Theorem 2.3** (DFS).**

Let $u$ be a viscosity solution to (1.2) satisfying

[TABLE]

There exists a constant $\bar{\epsilon}(a_{0}^{-},a_{1}^{-})$ such that if $\epsilon\leq\bar{\epsilon}$ then $\Gamma(u)$ is $C^{1,\alpha}$ in $B_{1/2}$, and the $C^{1,\alpha}$ norm of $\Gamma(u)$ is bounded by a universal constant.

We need a refinement of Theorem 2.3 in which the approximation of $u$ in a $C^{1,\alpha}$ fashion is done by using two-phase quadratic polynomials rather than two-plane solutions. For this we first introduce the family $V_{a^{\pm},M,\nu}$ of quadratic approximations with general slopes $a^{+},a^{-}\in[1/2,2]$ as

[TABLE]

with $\nu$ a unit direction and

[TABLE]

Notice that

[TABLE]

while

[TABLE]

Next we approximate $u$ by elements of the family $V$ with quadratic part $M$ of order $\epsilon^{1/2}\gg\epsilon$.

Proposition 2.4**.**

Assume that $0\in\Gamma(u),$ and $\mathcal{F}$ satisfies (1.3). There exists a universal constant $r_{1}>0$, such that if $u$ satisfies

[TABLE]

with

[TABLE]

for some $0<\epsilon\leq\epsilon_{1}$, then

[TABLE]

for some $\alpha_{1}$ universal, with

[TABLE]

and $C$ universal.

The proof of Proposition 2.4 is postponed till Section 5. We will follow the arguments in [DFS] where the same result was obtained without the presence of the quadratic part, i.e. in the case $M=0$, $\bar{M}=0$.

Notice that the conclusion of Proposition 2.4 can be iterated indefinitely. Indeed, if $u$ satisfies (2.7), then the rescaling

[TABLE]

will satisfy

[TABLE]

In order to apply Proposition 2.4 once again, we need to check that

[TABLE]

This is clearly satisfied since (by choosing $\epsilon_{1}$ possibly smaller)

[TABLE]

A consequence of Proposition 2.4 is the corollary below which is a refined version of Theorem 2.3.

Corollary 2.5**.**

There exist universal constants, $\epsilon_{1},\delta>0$ such that if (2.5)-(2.6) hold with $0<\epsilon\leq\epsilon_{1},$ then $\Gamma(u)$ has small $C^{1,\alpha}$ norm in $B_{1/2}$ and

[TABLE]

[TABLE]

with $C>0$ universal.

3. The proof of Theorem 1.1

In this section we will provide the proof of our main Theorem 1.1. In Lemma 3.1 we show that $u$ can be well-approximated by a two-plane solution with maximal slopes at some scale, possibly small. Then we use Proposition 1.3 to obtain a “reversed” improvement of flatness property and conclude that the solution $u$ is well approximated by the same two-plane solution at all large scales.

Let $u$ be a Lipschitz viscosity solution to (1.2) in $\mathbb{R}^{n}$. Call,

[TABLE]

The next lemma provides an initial flatness condition for $u$ at some arbitrary scale.

Lemma 3.1**.**

Assume that $a,b>0.$Then,

[TABLE]

Moreover, there exists a sequence $u_{k}(x):=\frac{u(x_{k}+d_{k}x)}{d_{k}}$ of rescalings of $u$ in $B_{2},$ with

[TABLE]

uniformly on compacts of $B_{2}.$

Proof.

Assume that

[TABLE]

Given $\epsilon>0$, let $x_{\epsilon}\in\{u>0\}$ be such that

[TABLE]

Let $d$ be the distance from $x_{\epsilon}$ to $\Gamma(u)$ and consider the rescalings

[TABLE]

These rescalings will still satisfy (1.2) say in $B_{2},$ with $B_{1}\subset B_{2}^{+}(u_{\epsilon})$ being tangent to $\Gamma(u_{\epsilon})$. In fact, after a rotation we can also assume that

[TABLE]

Since the $u_{\epsilon}$ are uniformly Lipschitz (up to extracting a subsequence) we can conclude that

[TABLE]

uniformly on compacts in $B_{2}$. In fact, since the $u_{\epsilon}$ are uniformly $C^{2,\alpha}$ in the interior, the convergence is in the $C^{2,\alpha}$ norm on compact subsets of $B^{+}_{2}(\bar{u})\cup B^{-}_{2}(\bar{u}).$

In particular, $\bar{u}$ solves (1.2) and it satisfies

[TABLE]

with $B_{1}\subset B_{2}^{+}(\bar{u})$ tangent to $\Gamma(\bar{u}).$ In fact $\bar{u}$ solves $\mathcal{F}(D^{2}\bar{u})=0$ in $B_{1}$ and $\bar{u}\geq 0$ in $B_{1}$ but it is not identically zero because $\bar{u}_{n}(0)=a>0$. Thus $\bar{u}>0$ in $B_{1}.$ Moreover, since all the $u_{\epsilon}$ vanish at a point on $\partial B_{1}$ it follows that $B_{1}$ is tangent to $\Gamma(\bar{u}).$

Now by Proposition 5.5 in [CC], $\bar{u}_{n}\in\mathcal{S}(\lambda,\Lambda)$, hence by the strong maximum principle

[TABLE]

where $\mathcal{O}$ is the connected component on $B_{2}^{+}(\bar{u})$ containing $B_{1}.$ On the other hand,

[TABLE]

hence

[TABLE]

Moreover $\bar{u}=0$ at the point where the unit ball is tangent to $F(\bar{u})$ thus

[TABLE]

Now, call (see (3.1) and recall $a,b>0$)

[TABLE]

Since $|\nabla\bar{u}|\leq b$ in $B_{2}^{-}(\bar{u})$ and (3.3) holds, the two-plane solution

[TABLE]

touches $\bar{u}$ by below on $B_{2}\cap\{x_{n}=-1\}$. This is possible only if $p\equiv\bar{u}$ and this contradicts that $|\nabla\bar{u}|\leq b$ in $B_{2}^{-}(\bar{u}).$

Similarly, if we assume that $a<G(b)$ we can argue as above, starting with a point $x_{\epsilon}\in\{u<0\}$ where $|\nabla u(x_{\epsilon})|>b-\epsilon$ and reach a contradiction. Thus, we have shown that

[TABLE]

and moreover

[TABLE]

which proves the second part of the lemma. ∎

Since $\mathcal{F}$ is homogeneous of degree 1, after multiplication by a constant we can assume without loss of generality that

[TABLE]

and this assumption will be made throughout the paper from now on. Then, the following corollary holds.

Corollary 3.2**.**

Assume $0\in\Gamma(u)$ and $te_{n}\in B_{2}^{+}(u)$ for all $t\in(0,1]$. If

[TABLE]

for some $\eta>0$, then

[TABLE]

with $\epsilon(\eta)\to 0$ as $\eta\to 0.$

Proof.

This follows immediately by compactness from Lemma 3.1.

Assume by contradiction that there exist a sequence $\eta_{k}\to 0$ and a sequence of equi-Lipschitz solutions $u_{k}$ to a sequence of problems with operators $\mathcal{F}_{k}\in\mathcal{E}(\lambda,\Lambda)$ and such that

[TABLE]

but

[TABLE]

Then, from $u_{k}(0)=0$ and (3.4) we conclude that

[TABLE]

Since $|\nabla u_{k}|\leq 1$ this implies that

[TABLE]

We can now argue as in the previous lemma and extract a subsequence which will converge uniformly on $\overline{B}_{2}$ to $\bar{u}=x_{n}$ (since $a=b=1$) and contradict (3.5).

∎

We are now ready to prove Theorem 1.1, assuming that Proposition 1.3 holds.

First, we remark that if alternative $(i)$ of Proposition 1.3 is satisfied, then we can rescale and iterate one more time. Precisely, if $u$ satisfies alternative $(i)$, then the rescaling

[TABLE]

will satisfy

[TABLE]

In order to apply Proposition 1.3 once again, we need to check that

[TABLE]

This is clearly satisfied since (for $\epsilon_{0}$ small enough)

[TABLE]

Proof of Theorem $\ref{T2}.$ Let $\epsilon_{0}$ be a sufficiently small universal constant so that the conclusions of Proposition 1.3, Proposition 2.4 and Corollary 2.5 hold for all $\epsilon\leq\epsilon_{0}$, and let $\epsilon(\eta)$ be as in Corollary 3.2.

According to Lemma 3.1, after a translation, a rotation and a rescaling, we can assume that

[TABLE]

for $\epsilon^{\prime}$ small to be made precise later. If $\epsilon^{\prime}\leq\epsilon_{0}$, then $\Gamma(u)$ is locally $C^{1,\alpha}$ and $u$ is a classical solution by Corollary 2.5.

By iterating Proposition 2.4 (say with $M\equiv 0$) and by interior $C^{1,\alpha}$ estimates for fully nonlinear equations it follows that,

[TABLE]

and

[TABLE]

for some $C$ universal.

We choose $\bar{\eta}$ universal, small enough so that

[TABLE]

and we claim that

Lemma 3.3**.**

[TABLE]

provided that $\epsilon^{\prime}$ is chosen sufficiently small.

Then, according to Corollary 3.2,

[TABLE]

for all $R$ large. This, combined with the improvement of flatness Proposition 2.4 (say with $M\equiv 0$) implies that $u$ must be a two-plane solution and concludes the proof of our main Theorem 1.1.

We are left with the proof of Lemma 3.3.

Proof of Lemma 3.3. Denote $C\epsilon^{\prime}$ from (3.7) by $\epsilon^{\prime\prime}:=C\epsilon^{\prime}$ and we will choose $\epsilon^{\prime\prime}$ (and therefore $\epsilon^{\prime}$) later, so that $\epsilon^{\prime\prime}\ll\bar{\eta}$.

Let $\bar{t}$ be the first $t$ for which (3.9) fails, and assume without loss of generality after a rescaling that $\bar{t}=1$, i.e.

[TABLE]

Notice that after rescaling (3.8) is still satisfied, i.e.

[TABLE]

From Corollary 3.2 and (3.11), we conclude that

[TABLE]

Corollary 2.5 gives that $\Gamma(u)\cap B_{1}$ is a $C^{1,\alpha}$ graph in the $e_{n}$ direction with small norm.

Since $1-u_{n}\geq 0$ belongs to the class of solutions to linear equations with measurable coefficients $\mathcal{S}(\lambda,\Lambda)$ in $\{u>0\}$, the Harnack inequality and $(1-u_{n})(e_{n})=\bar{\eta}$ give that

[TABLE]

In $B_{1}^{+}(u)\setminus B_{1/4}(e_{n}/2)$ we compare $1-u_{n}$ with the solution to $\mathcal{M}^{+}(D^{2}w)=0$ which equals zero on $\partial B_{1}^{+}(u)$ and $c\bar{\eta}$ on $\partial B_{1/4}(\frac{1}{2}e_{n})$. Using the Hopf lemma in $C^{1,\alpha}$ domains together with $C^{1,\alpha}$ estimates up to the boundary we conclude that $w$ grows linearly away from $\Gamma(u)$ hence

[TABLE]

Integrating in the $e_{n}$ direction and using $u(0)=0$, and $c$, $\bar{\eta}$ are universal we find

[TABLE]

for some small universal constant $c_{1}>0$.

From (3.12) we have

[TABLE]

We now apply Proposition 1.3 and conclude that either alternative $(i)$ or $(ii)$ is satisfied. However if $\epsilon^{\prime\prime}$ is small enough, then alternative $(ii)$ cannot hold since otherwise by (3.8)

[TABLE]

and we reach a contradiction. Similarly, after applying the conclusion of Proposition 1.3 a number of $N$ times, we obtain that if $\epsilon^{\prime\prime}$ is sufficiently small depending on $N$ and $\epsilon_{0}$, $c_{0}$, $r_{0}$, then only alternative $(i)$ can hold for the $N$-iterations and conclude that

[TABLE]

with

[TABLE]

Here we also used Corollary 2.5 (see (2.9)). Now notice that,

[TABLE]

which implies

[TABLE]

Thus, if $\epsilon^{\prime\prime}$ is small enough depending on $r$, the inequality above together with (3.16) gives that

[TABLE]

Thus, since $M\nu=0$ and $\|M\|\leq 1,$

[TABLE]

which combined with (3.15) gives that

[TABLE]

This contradicts (3.14) for $t=r/2$, as long as $r$ is small enough universal, i.e. $N$ is large enough and $\epsilon^{\prime\prime}$ is small enough. ∎

4. The transmission problem

In this section we study properties of solutions to the nonlinear transmission problem (4.1) below. This type of transmission problem appears as the linearization to the free boundary problem (1.2) and our goal is to obtain a version of the key Proposition 1.3 in this linearized setting. In Section 5 we will use compactness methods and extend the result to the nonlinear setting.

Consider the transmission problem

[TABLE]

with $0<b_{0}\leq b\leq b_{1}$ and say $\|v\|_{\infty}\leq 1.$ Here $\mathcal{F}\in\mathcal{E}(\lambda,\Lambda)$.

By abuse of notation, in this section we denote by $v^{\pm}$ the restrictions of $v$ to $B_{1}\cap\{x_{n}\geq 0\}$ and respectively $B_{1}\cap\{x_{n}\leq 0\}.$ Call,

[TABLE]

In [DFS] the authors showed that solutions to the transmission problem above belong to the class $\mathcal{C}^{1,\alpha}(B_{1}^{\pm})$, for some universal $\alpha$. We recall here the definition of viscosity solutions to the problem (4.1) and the precise result from [DFS]. Constants depending on $n,\lambda,\Lambda,b_{0},b_{1}$ are called universal.

Definition 4.1**.**

We say that $v\in C(B_{1})$ is a viscosity subsolution (resp. supersolution) to (4.1) if

(i) \mathcal{F}(D^{2}v^{\pm})\geq 0\ (\text{resp. \leq 0})\quad\text{in B^{\pm}_{1}}, in the viscosity sense;

(ii) $v_{n}^{+}\geq b\,v_{n}^{-}$ on $B_{1}\cap\{x_{n}=0\}$ in the viscosity sense which means that if

[TABLE]

touches $v$ by above (resp. by below) at $x_{0}\in\{x_{n}=0\}$ for some quadratic polynomial $P(x^{\prime})$ in $x^{\prime}$, then

[TABLE]

If $v$ is both a viscosity subsolution and supersolution to (4.1), we say that $v$ is a viscosity solution to (4.1).

The next result is contained in Theorems 3.2 and 3.3 in [DFS].

Theorem 4.2** (DFS).**

Let $v$ be a viscosity solution to (4.1) such that $\|v\|_{\infty}\leq 1$. Then $u\in\mathcal{C}^{1,\alpha}(B_{1/2}^{\pm})$ with a universal bound on the norms. In particular, there exists a universal constant $C$ such that

[TABLE]

for all $r\leq 1/4$ and with

[TABLE]

We also recall the definitions from [DFS] of the general classes of subsolutions $\underline{\mathcal{S}^{*}}$, supersolutions $\overline{\mathcal{S}^{*}}$ and solutions ${\mathcal{S}^{*}}$ suited for the transmission problem (4.1). In what follows,

[TABLE]

We denote by $\underline{\mathcal{S^{*}}}(\lambda,\Lambda)$ the class of functions $w\in C(B_{1})$ such that

[TABLE]

Analogously, $\overline{\mathcal{S^{*}}}(\lambda,\Lambda)$ denotes the class of functions $w\in C(B_{1})$ such that

[TABLE]

and

[TABLE]

For simplicity of notation we will drop the dependence on $\lambda,\Lambda$.

We restrict our attention to the case of concave operators.

Lemma 4.3**.**

Let $v$ be a viscosity solution of (4.1) and assume $\mathcal{F}$ is concave. Then the tangential second order quotients $v^{h}_{\tau\tau}$ are subsolutions, i.e.

[TABLE]

Here $\tau$ is a unit direction with $\tau\perp e_{n}$, and $h>0$.

Proof.

The fact that $v^{h}_{\tau\tau}$ is a subsolution in $B_{1}\setminus L$ is standard. By Theorem 4.2 $v^{h}_{\tau\tau}\in\mathcal{C}^{1,\alpha}(B_{1}^{\pm})$, hence the transmission condition on $L$ is satisfied in the classical sense and therefore it holds also in the viscosity sense. ∎

In Lemma 3.5 in [DFS] it was shown by the use of an explicit barrier that the Harnack inequality for the class $\mathcal{S}^{*}$ follows from the standard Harnack inequality.

Next we state the weak Harnack inequality for the classes $\overline{\mathcal{S}^{*}}$ and $\underline{\mathcal{S}^{*}}$.

Lemma 4.4**.**

Let $w\in\overline{\mathcal{S}^{*}}$ in $B_{1}$, $w\geq 0$ and $inf_{B_{1/2}}w\leq 1.$ There exist universal constants $0<\mu<1$ and $M>1$ such that

[TABLE]

Proof.

The proof follows from the standard weak Harnack inequality once we know that $w\leq C^{\prime}$ at some point in the set $\{|x^{\prime}|\leq 3/4,\quad|x_{n}|\geq\delta\}\cap B_{3/4}$. Otherwise we can use comparison principle in the cylinder $\mathcal{C}_{0}:=\{|x^{\prime}|\leq 3/4,\quad|x_{n}|\leq\delta\}$ and conclude that

[TABLE]

provided that $\delta$ is small. This implies that $w>1$ in $B_{1/2}\cap\mathcal{C}_{0}$ and we contradict the hypothesis that $inf_{B_{1/2}}w\leq 1.$ ∎

As in the proof of Theorem 4.8 in [CC], we can iterate the lemma above and the standard weak Harnack inequality at smaller scales and obtain the version of weak Harnack inequality for subsolutions (here $w_{+}$ denotes the positive part of $w$.)

Theorem 4.5** (Weak Harnack).**

Let $w\in\underline{\mathcal{S^{*}}}$ in $B_{1}$. Then, for any $p>0,$

[TABLE]

with $C(p)$ depending on $p$ and the universal constants.

As mentioned in the Introduction, the $C^{2,\alpha}$ estimates play an important role in our analysis, however the Evans-Krylov theorem for the transmission problem (4.1) is not known. In the next proposition we show that, if in addition the solution $v$ is monotone decreasing in the $e_{n}$ direction, then either there is indeed a pointwise $C^{2,\alpha}$ type estimate at [math] or the $e_{n}$ derivative of $v$ is strictly negative.

We assume that

[TABLE]

We wish to prove the following Harnack type inequality for $v_{n}^{\pm}.$

Proposition 4.6** (Linear dichotomy).**

Let $v$ be as above and suppose that

[TABLE]

Then, there exist universal constants $c_{0},r_{0}>0$ such that either

1. (i)

[TABLE]

or 2. (ii)

[TABLE]

where $Q(x^{\prime})$ is a quadratic polynomial in the $x^{\prime}$ direction with $\|Q\|\leq C$ universal and

[TABLE]

Proof.

Let $r_{0}$ be given, to be specified later. Now, assume by contradiction that we can find a sequence $c_{k}\to 0$ and sequences of convex operators $\mathcal{F}_{k}\in\mathcal{E}(\lambda,\Lambda),$ constants $b_{0}\leq b_{k}\leq b_{1}$ and bounded monotone solutions $v_{k}$ to (4.1) which satisfy

[TABLE]

and for which (ii) does not hold. Then, up to extracting a subsequence, $v_{k}$ converges uniformly on compacts, and in the $C^{1,\alpha}$-norm from either side of $L$, to a solution $\bar{v}$ for a limiting problem

[TABLE]

with

[TABLE]

and

[TABLE]

Also, $\mathcal{\bar{F}}\in\mathcal{E}(\lambda,\Lambda)$, $\mathcal{\bar{F}}(0)=0$ and $\mathcal{\bar{F}}$ is concave.

After subtracting a linear function in the $x^{\prime}$ variable we can assume that (in view of the free boundary condition $\bar{v}_{n}^{-}(0)=0,$)

[TABLE]

Step 1. We prove that

[TABLE]

or in other words that $\bar{v}$ is uniformly semiconcave in the $x^{\prime}$ variable.

For this it suffices to show that

[TABLE]

independently of $h$, where $\bar{v}_{\tau\tau}^{h}$ are the tangent second difference quotient as in Lemma 4.3. We claim that

[TABLE]

which combined with Lemma 4.3 and the weak Harnack inequality (4.4) for small $p>0$ implies the desired bound. To prove (4.13), notice that by the Evans-Krylov $C^{2,\alpha}$ interior estimates in $B_{|x_{n}|/2}(x)$ we have

[TABLE]

Since we can write,

[TABLE]

(4.13) follows. ∎

Step 2. In this step we wish to show that

[TABLE]

with $C$ universal. Below, the constant $C$ may change from line to line.

Set

[TABLE]

Then $\tilde{v}$ satisfies (4.9)-(4.10). From (4.11) we also have,

[TABLE]

Then the conclusion of Step 1 implies that

[TABLE]

Hence, by the monotonicity of $\tilde{v}$,

[TABLE]

Next we claim that

[TABLE]

for some large constant $M$ to be specified later. Suppose by contradiction that this does not hold. Then by Harnack inequality for $C-\tilde{v}\geq 0$ in $B_{1}^{+}$,

[TABLE]

with $K$ universal. In particular, if

[TABLE]

then,

[TABLE]

We compare $\tilde{u}$ with the explicit barrier:

[TABLE]

where $A=A(C,\lambda,\Lambda)$ is chosen so that

[TABLE]

Call

[TABLE]

We show that for $d$ small enough (hence $M$ large enough)

[TABLE]

Thus, we conclude that the inequality holds in $R$ and we reach a contradiction because,

[TABLE]

Now we check (4.20). On $\{x_{n}=d\}$ this follows from (4.19), if $d$ is chosen small depending on $C$ and $A.$ Similarly, on $\{x_{n}=0\}$ the desired bound follows immediately from (4.16). Finally in the set $\{0<x_{n}<d,|x^{\prime}|=1/3\}$ we use (4.17) and again we obtain (4.20) for $d$ sufficiently small. In conclusion the claim (4.18) holds.

Using the $\tilde{v}$ in decreasing in the $e_{n}$ we obtain (after relabeling $C$)

[TABLE]

and recall that $\tilde{v}(0)=0.$ We apply Harnack inequality for $\tilde{v}+C\in\mathcal{S}^{*}$ (see Lemma 3.4 in [DFS]) and conclude that

[TABLE]

which after rescaling gives the desired claim (4.14).

Step 3. We prove that if $\bar{v}$ solves (4.9) and satisfies (4.10)-(4.11)-(4.14), then there exists a universal $r_{0}$ such that alternative (ii) holds for $\bar{v}$ with right hand side $\frac{1}{8}r_{0}^{2}$ instead of $\frac{1}{4}r_{0}^{2}$.

This claim follows by compactness from the classification of global solution obtained in Lemma 4.7 below.

Assume by contradiction that there exists a sequence of $\delta_{k}\to 0$ and of solutions $\bar{v}_{k}$ to a sequence of problems $\eqref{tra_limit}_{k}$ (satisfying the same properties as $\bar{v}$) for which the alternative (ii) fails in the ball of radius $\delta_{k}.$ Denote the quadratic rescalings by

[TABLE]

Then, up to extracting a subsequence, $w_{k}$ converges uniformly on compacts to a global solution $U$ to a limiting transmission problem ($\mathcal{G}\in\mathcal{E}(\lambda,\Lambda),b_{0}\leq g\leq b_{1},$)

[TABLE]

with the convergence being in the $C^{1,\alpha}$ norm from either side of $L$, up to $L$, and in the $C^{2,\alpha}$ norm in the interior. Clearly the global solution $U$ also satisfies (4.10)-(4.11)-(4.14), and the operator $\mathcal{G}$ is concave as the limit of the corresponding $\mathcal{\bar{F}}_{k}$.

Then according to Lemma 4.7 below, we conclude that $U=Q(x^{\prime})$ with $Q(x^{\prime})$ a pure quadratic polynomial in the $x^{\prime}$-direction and with $\mathcal{G}(D^{2}Q)=0$. Therefore, for $k$ large, $\bar{v}_{k}$ satisfies the alternative (ii) in $B_{\delta_{k}}$ with $Q_{k}:=Q+t_{k}|x^{\prime}|^{2}$ for an appropriate choice of $t_{k}$’s, so that $\mathcal{\bar{F}}_{k}(D^{2}Q_{k})=0$ and $t_{k}\to 0$ as $k\to\infty$. We reached a contradiction and therefore we have established Step 3.

End of the proof: By Step 3, $\bar{v}$ satisfies alternative (ii) with right hand side $\frac{1}{8}r_{0}^{2}$ for some quadratic polynomial $\bar{Q}(x^{\prime})$ with $\mathcal{\bar{F}}(D^{2}\bar{Q})=0$. As above, this means that the $v_{k}$’s satisfy the alternative (ii) for all $k$ large, which is a contradiction.

∎

Lemma 4.7**.**

Let $U$ be a global solution to

[TABLE]

with $\mathcal{G}\in\mathcal{E}(\lambda,\Lambda)$, $\mathcal{G}$ concave, $0<b_{0}\leq g\leq b_{1}$, and

[TABLE]

for some constant $C$. Then $U=Q(x^{\prime})$ with $Q(x^{\prime})$ a pure quadratic polynomial in the $x^{\prime}$-direction, such that $\mathcal{G}(D^{2}Q)=0.$

Proof.

Let

[TABLE]

be a sequence of blow-downs which converges uniformly on compacts to another global solution $\bar{U}$ to (4.22)-(4.23).

Let $\tau$ be a unit tangential direction, $\tau\perp e_{n}$ and denote by,

[TABLE]

Notice that $\alpha\neq\infty$. Indeed, from Step 1 in the proof of Proposition 4.6 we obtain that $\partial_{\tau\tau}U^{h}_{k}$ is bounded above in $B_{1}$ independent of $h$, and this implies that $\gamma$ is well defined.

We claim that

[TABLE]

Let $x_{0}$ be a point, say for simplicity in $B_{1}\setminus L$, and let us show that

[TABLE]

Since $U_{k}\to\bar{U}$ in $C^{2,\alpha}$ in a small ball $B\subset\{x_{n}\neq 0\}$ around $x_{0}$, clearly,

[TABLE]

Assume by contradiction that the inequality above is strict. Then for all $k$ large,

[TABLE]

for some small $\delta>0.$ By $C^{2,\alpha}$ regularity,

[TABLE]

hence, for all $h>0$ small

[TABLE]

On the other hand form the definition of $\gamma$ we have

[TABLE]

Since $(U^{h}_{k})_{\tau\tau}\in\underline{\mathcal{S}^{*}}$ we can construct an explicit upper barrier in $B_{3/4}\setminus B_{c^{\prime}}(x_{0})$ (see Lemma 3.4 in [DFS]), and conclude that

[TABLE]

for all $k$ large. This contradicts the definition of $\gamma,$ and the claim (4.24) is proved.

Next we show that $\bar{U}$ equals a quadratic polynomial $Q^{+}$ (resp. $Q^{-}$) in $\{x_{n}>0\}$ (resp. $\{x_{n}<0\}$) and

[TABLE]

We know that

[TABLE]

hence

[TABLE]

for some quadratic polynomial $Q$ and functions $b$, $a_{i}$ depending on one variable.

In the set $\{x_{n}>0\}$ we have for $i<n$

[TABLE]

where we have used that $Q_{i}$ is a linear function. This means that $a_{i}$ is linear in the set $\{x_{n}>0\}$. Now we use $\bar{U}_{n}\in\mathcal{S}$ and we argue as above to find that $b^{\prime}$ is linear in $\{x_{n}>0\}$ and our claim (4.25) is proved.

Moreover,

[TABLE]

Finally, since $Q^{\pm}$ both solve (from either side of $L$),

[TABLE]

we conclude that

[TABLE]

and therefore

[TABLE]

Let us call this common polynomial $Q.$ In particular, since $Q_{n}\leq 0$ it follows that $Q_{n}\equiv 0$ . Thus, $Q$ is a pure quadratic polynomial in the $x^{\prime}$ direction.

Finally, we need to show that $U\equiv Q.$

First notice that, for all tangential directions $\tau$,

[TABLE]

Thus, $U-Q$ is concave in the $x^{\prime}$-direction. Moreover,

[TABLE]

Hence,

[TABLE]

and by the monotonicity of $U$ we get

[TABLE]

Since $U-Q\in\mathcal{S}$, by (4.26) and Hopf lemma we conclude that $U=Q$ in $\{x_{n}\geq 0\}$. By the monotonicity of $U$ we have $U\geq Q$ in the set $\{x_{n}\leq 0\}$, hence $U^{-}=Q$ again by Hopf lemma. ∎

5. The free boundary problem

In this section we prove Proposition 2.4 and Proposition 1.3.

Let $u$ be a Lipschitz continuous viscosity solution to (1.2) in $\mathbb{R}^{n}$. Recall that, following the arguments in Section 3, after an initial dilation we can assume without loss of generality that (see (3.1))

[TABLE]

hence $|\nabla u|\leq 1.$

5.1. Notation

As in the Introduction, we denote by $P_{M,\nu}$ the quadratic polynomial

[TABLE]

with

[TABLE]

and by $V_{a^{\pm},M,\nu}$ the two-phase quadratic polynomial of slopes $a^{+},a^{-}\in[\frac{1}{2},2]$

[TABLE]

Given a continuous function $v$ in say $B_{1}$, we denote its $\epsilon$-linearization around the function $V_{a^{\pm},M,\nu}$ above as

[TABLE]

In what follows, we will typically drop the indices from $V,P,\tilde{v}$ whenever there is no possibility of confusion.

Remark 5.1*.*

We remark that if $v,w\in C(B_{1})$ and $v\geq w$ in $B_{1}$ then $\tilde{v}\geq\tilde{w}$ in $B_{1}.$ This claim is obvious if $v,w$ have the same sign. If $v\geq 0>w$, then

[TABLE]

We now want to construct appropriate $\epsilon$-perturbations of $V$ which are strict subsolutions to the two-phase problem in say $B_{1}.$ Similarly, one can construct strict supersolutions.

Given $\epsilon,\delta>0,$ assume that

[TABLE]

Let $N$ be a $n\times n$ diagonal matrix such that

[TABLE]

and let $p,q,A\in\mathbb{R}$ and $\xi^{\prime}\in\mathbb{R}^{n-1}.$ Given $\epsilon>0$, set

[TABLE]

[TABLE]

and call,

[TABLE]

Lemma 5.2**.**

There exists a constant $k$ depending on $p,q,N,\xi^{\prime}$ and the universal parameters, such that if $p,q$ satisfy

[TABLE]

then $v$ defined in (5.2) is a strict subsolution to (1.2) in $B_{1}$ for all $\epsilon$ sufficiently small.

Proof.

In $B_{1}^{\pm}(v)$ since $\mathcal{F}$ is homogeneous of degree 1 and $\mathcal{F}(M)=0$

[TABLE]

For the free boundary condition, let us compute on $\Gamma(v)$ (dependence of constants on the parameters is not explicitly noted, and constants may change from line without being renamed)

[TABLE]

We use that $|\bar{M}|\leq 2\delta\epsilon^{1/2}$ and that $(Nx)\cdot e_{n}=O(\epsilon^{1/2})$ on $\Gamma(v)$ , hence

[TABLE]

and similarly

[TABLE]

Thus,

[TABLE]

[TABLE]

as long as $\epsilon$ is small enough (depending on $p,q,N,\xi^{\prime},\delta$) and for the appropriate choice of $k$. ∎

Remark 5.3*.*

In the inequality above it suffices only to assume that the modulus of continuity $\omega$ of $G^{\prime}$ satisfies $\omega(0+)\leq\delta^{2}$.

Remark 5.4*.*

In particular, since the $v_{t}(x):=v(x+te_{n})$ form a continuous family of subsolutions with $t\to-\infty$, we conclude that $u$ satisfies the comparison principle with translates of $v$. Hence, by Remark 5.1, $\tilde{u}$ and $\tilde{v}_{t}$ also satisfy the comparison principle. It is also easily seen that $\tilde{v}_{\epsilon t}$ converges locally uniformly as $\epsilon\to 0$ to the following function

[TABLE]

We will use this fact in the next subsection.

5.2. $C^{1,\alpha}$ estimates

In this subsection, we prove Proposition 2.4. Arguing as in [DFS], we first establish the following Harnack type inequality for $u$.

Lemma 5.5**.**

There exist universal constant $\bar{\epsilon},\delta>0$ such that if $u$ satisfies

[TABLE]

with

[TABLE]

and at $\bar{x}=\dfrac{1}{5}e_{n}$

[TABLE]

then

[TABLE]

for some $0<c<1$ universal.

Proof.

The proof follows the lines of Lemma 4.3 in [DFS]. For the reader convenience we sketch the main details.

We chose a specific function of the form $w$ from Remark 5.4. Precisely, let

[TABLE]

that is

[TABLE]

and with

[TABLE]

Call, for $\gamma>0,$

[TABLE]

One can find $\gamma,\eta,c>0$ small universal, so that

[TABLE]

and

[TABLE]

Now, let

[TABLE]

with $c_{1}$ to be specified later and call $v$ the function as in (5.2), associated to our choice of $A,\xi^{\prime},p,q,N.$ In view of Proposition 5.2, $v$ is a strict subsolution provided that $\delta$ is chosen sufficiently small. From Remark 5.4, we conclude that $\tilde{v}_{\frac{c_{1}}{8}\epsilon}$ and $\tilde{u}$ satisfy the comparison principle, with $\tilde{v}_{\frac{c_{1}}{8}\epsilon}$ converging uniformly to $w$, as $\epsilon\to 0.$

In particular, for $\epsilon$ small,

[TABLE]

and

[TABLE]

It follows that if

[TABLE]

then the inequality holds in $R_{\alpha}$ as well, and in particular

[TABLE]

From this, the desired claim immediately follows in $B_{\eta}$. A standard covering argument gives the claim in the full $B_{1/2}.$

We are left with the proof of (5.9). From assumption (5.4) and (5.8) we have that

[TABLE]

Moreover, by Harnack inequality and assumption (5.6), one can guarantee that

[TABLE]

for some $c_{0}(\alpha)$ universal. Hence,

[TABLE]

if $c_{1}$ is chosen appropriately. Again, from the uniform convergence of $\tilde{v}_{\frac{c_{1}}{8}\epsilon}$ we obtain

[TABLE]

and our claim is proved.

∎

Proof of Proposition $\ref{c1}.$ The proof follows the line of Lemma 5.1 in [DFS]. For the reader convenience we sketch the details.

We divide the proof into 3 steps.

Step 1 – Compactness. Fix $r_{1}$ universal to be made precise later in Step 3. Assume by contradiction that we can find sequences $\epsilon_{k}\rightarrow 0,\delta_{k}\to 0$ and a sequence $u_{k}$ of solutions to (1.2) in $B_{1}$ for a sequence of operators $\mathcal{F}_{k}\in\mathcal{E}(\lambda,\Lambda)$ and free boundary conditions $G_{k}$ such that

[TABLE]

with

[TABLE]

but $u_{k}$ does not satisfy the conclusion (2.7) of the proposition.

Let $\tilde{u}_{k}:=\tilde{u}_{a^{\pm}_{k},M_{k},e_{n},\epsilon_{k}}$ be defined as in (5.1). Then (5.12) gives,

[TABLE]

From Lemma 5.5 we obtain that the oscillation of $\tilde{u}$ decreases by a factor $1-c$ as we restrict from $B_{1}$ to $B_{1/2}$. This result can be iterated $m$-times provided that $\epsilon_{k}(2(1-c))^{m}\leq\bar{\epsilon}$. By Ascoli-Arzela theorem it follows that, as $\epsilon_{k}\rightarrow 0$, $\tilde{u}_{k}$ (up to a subsequence) converge uniformly in $B_{1/2}$ to a Hölder continuous function $u^{*}$. Also, up to a subsequence

[TABLE]

and hence

[TABLE]

Notice that $0\in\Gamma(u_{k})$ implies $\tilde{u}_{k}(0)=0$ hence $u^{*}(0)=0$.

Step 2 – Limiting Solution. We now show that $u^{*}$ solves

[TABLE]

with ${\mathcal{F}}^{*}\in\mathcal{E}(\lambda,\Lambda)$ concave, and $a=\bar{a}^{+},b=\bar{a}^{-}{G^{*}}^{\prime}(\bar{a}^{-}).$

Set,

[TABLE]

Then (recall that $\mathcal{F}_{k}(M_{k})=0$), ${\mathcal{F}}^{*}_{k}\in\mathcal{E}(\lambda,\Lambda)$ and it is concave. Thus, up to extracting a subsequence,

[TABLE]

Moreover, since $\mathcal{F}_{k}$ is homogeneous of degree 1,

[TABLE]

and similarly ${\mathcal{F}}^{*}_{k}(D^{2}\tilde{u}_{k})=0$ in $B_{1}^{-}(u_{k})$. Then, by standard arguments (see Proposition 2.9 in [CC]), we conclude that

[TABLE]

Next we verify the transmission condition in the viscosity sense of Definition 4.1. Let

[TABLE]

with $A\in\mathbb{R}$, $N$ a diagonal matrix with $\mathcal{M}^{-}(N)>0,$ and

[TABLE]

Assume that $w$ touches $u^{*}$ strictly by below at a point $x_{0}=(x_{0}^{\prime},0)\in B_{1/2}.$ Set,

[TABLE]

[TABLE]

and

[TABLE]

Recall that $\tilde{V}_{k}$ converges uniformly to $w$ on $B_{1/2}$ (see Remark 5.4). Since $\tilde{u}_{k}$ converges uniformly to $u^{*}$ and $w$ touches $u^{*}$ strictly by below at $x_{0}$, we conclude that for a sequence of constants $c_{k}\to 0$ and points $x_{k}\to x_{0}$ the function

[TABLE]

touches $u_{k}$ by below at $x_{k}$. Proposition 5.2 gives that $w_{k}$ is a strict subsolution to our free boundary problem, provided that we first choose $\delta$ small enough, so to guarantee that (5.3) holds. We reach a contradiction as we let $k\to\infty$, thus $u^{*}$ is a solution to the linearized problem (5.14).

Step 3 – Contradiction. Since $\tilde{u}_{k}$ converges uniformly to $u^{*}$ and $u^{*}$ enjoys the $C^{1,\alpha}$ estimate of Theorem 4.2 we have

[TABLE]

with

[TABLE]

Call,

[TABLE]

and

[TABLE]

We claim that we can decompose

[TABLE]

with

[TABLE]

hence $\|D_{k}\|=O(\epsilon_{k}^{3/2}).$

Indeed, since $M_{k}e_{n}=0$ first we can decompose

[TABLE]

such that $M_{k}\nu_{k}=0,$ and

[TABLE]

Since $\mathcal{F}_{k}(M_{k})=0$ we obtain that

[TABLE]

Then, using ellipticity we can decompose $\tilde{M}_{k}$ further, (here $\nu_{k}^{\perp}$ is a unit vector perpendicular to $\nu_{k}$)

[TABLE]

so that

[TABLE]

and the claim (5.17) is proved.

Let us show now that for $r=r_{1}$ universal, (say $\alpha_{1}=\alpha/2$)

[TABLE]

which contradicts the fact that $u_{k}$ does not satisfy (2.7). From (5.16) and the definition of $\tilde{u}_{k}$ we get that in $B_{r}\cap(B_{1}^{+}(u_{k})\cup\Gamma(u_{k}))$ (we can argue similarly in the negative part)

[TABLE]

Since in this set $x_{n}\geq-3\epsilon_{k}^{1/2}$ we conclude that for all $k$ large

[TABLE]

which gives

[TABLE]

Finally, from (5.17) we conclude that

[TABLE]

from which the desired bound follows for $r_{1}$ small enough. ∎

5.3. Nonlinear Dichotomy

In this subsection, we prove the Proposition 1.3. The proof is very similar to the one of Proposition 2.4 above except that we use Proposition 4.6 for the limiting transmission problem.

Proof.

Step 1 – Compactness and Limiting solution. Fix $r_{0}$ universal to be specified later. Assume by contradiction that we can find sequences $\epsilon_{k}\rightarrow 0,\delta_{k}\to 0$ and a sequence $u_{k}$ of solutions to (1.2) in $B_{1}$ for a sequence of operators $\mathcal{F}_{k}\in\mathcal{E}(\lambda,\Lambda)$ and free boundary conditions $G_{k}$ such that

[TABLE]

with

[TABLE]

but $u_{k}$ does not satisfy either of the alternatives (1.10) (for some small constant $c_{0}^{\prime}$ to be specified later) or (1.11) .

Let $\tilde{u}_{k}:=\tilde{u}_{a^{\pm}_{k},M_{k},e_{n},\epsilon_{k}}$ be defined as in (5.1). In Step 1- Step 2 of the proof of Proposition 2.4 we showed that as $\epsilon_{k}\rightarrow 0$, $\tilde{u}_{k}$ converge uniformly (up to a subsequence) in $B_{1/2}$ to a Hölder continuous function $u^{*}$ with $u^{*}(0)=0$, $\|u\|_{L^{\infty}}\leq C$ and $u^{*}$ solves the transmission problem

[TABLE]

with ${\mathcal{F}}^{*}\in\mathcal{E}(\lambda,\Lambda)$ concave, and $b>0$ bounded by universal constants.

Next we show that $u^{*}$ is monotone decreasing in the $e_{n}$ direction,

[TABLE]

Indeed, $\tilde{u}_{k}$ satisfies $\mathcal{F}^{*}_{k}(D^{2}\tilde{u}_{k})=0$ in $B_{1}^{+}(u_{k})$ and $\Gamma(u_{k})\subset\{|x_{n}|\leq\epsilon_{k}^{1/2}\}.$ In particular, if $\bar{x}\in\{x_{n}>0\}$, for $k$ large enough

[TABLE]

and

[TABLE]

Hence,

[TABLE]

Using that $|\nabla u_{k}(\bar{x})|^{2}\leq 1$ and $Me_{n}=0$ we get

[TABLE]

where the constant is $O(\epsilon_{k}^{3/2})$ depends on $\bar{x}_{n}.$ By $C^{1,\alpha}$ estimates the $\tilde{u}_{k}$’s converge to $u^{*}$ in $C^{1}$ in the ball $B$. Passing to the limit as $k\to\infty$, we get that

[TABLE]

By continuity, since $u^{*}$ is $C^{1,\alpha}$ up to $\{x_{n}=0\}$ we conclude that

[TABLE]

We argue similarly for $(u^{*}_{n})^{-}.$

Step 2 – Contradiction. According to Proposition 4.6 (since (5.19) holds and $u^{*}$ is bounded by a universal constant) there exist universal constants $c_{0},r_{0}>0$ such that either of the following alternative is satisfied:

1. (i)

[TABLE] 2. (ii)

[TABLE]

where $Q(x^{\prime})$ is a quadratic polynomial in the $x^{\prime}$ direction with $\|Q\|\leq C$ universal and

[TABLE]

Since $u^{*}(0)=0$ we have

[TABLE]

If (5.20) is satisfied, by the $C^{1,\alpha}$ estimates for $u^{*}$ we obtain that

[TABLE]

with $|p|,|q|,|\xi^{\prime}|\leq C$ and

[TABLE]

Using the convergence of the $\tilde{u}_{k}$ to $u^{*}$ and the definition of $\tilde{u}_{k}$ we immediately get that for $k$ large,

[TABLE]

Arguing as in Step 3 of the proof of Proposition 2.4 (using the same notation) we conclude that for $r\leq r_{1}$,

[TABLE]

Thus, after rescaling (see the argument after the statement of Proposition 2.4), Corollary 2.5 gives that

[TABLE]

provided that $r$ is chosen small, universal. Thus $u_{k}$ satisfies alternative (1.10) and we have reached a contradiction.

If (5.21) is satisfied, using the convergence of the $\tilde{u}_{k}$ to $u^{*}$ and the definition of $\tilde{u}_{k}$ we get

[TABLE]

with

[TABLE]

where the last equality follows from (see (5.15))

[TABLE]

As before denote

[TABLE]

We argue as in Step 3 of Proposition 2.4 and decompose

[TABLE]

with

[TABLE]

Thus, (5.23) yields for $k$ large,

[TABLE]

for $\alpha_{0}$ small enough, and again we reached a contradiction.

∎

5.4. Proof of Theorem 1.2

Here we only sketch the proof and we outline the differences between the case of quasilinear equations

[TABLE]

and the fully nonlinear concave equations case treated in Theorem 1.1.

We need to check that Lemma 3.1, Corollary 3.2, and Propositions 1.3 and 2.4 apply to the quasilinear setting.

The proofs of Lemma 3.1 and Corollary 3.2 are identical since Lipschitz solutions to (5.24) in $B_{1}$ satisfy interior $C^{2}$ estimates. This means that in any compact set of $B_{1}$, directional derivatives $u_{e}$ solve linear elliptic equations (with first order terms) and bounded coefficients.

For the proofs of Propositions 1.3 and 2.4 we introduce the approximate quadratic polynomials in this setting. Let

[TABLE]

such that $M$ and $\nu$ satisfy the compatibility conditions

[TABLE]

Also let $V_{a^{\pm},M,\nu}$ and $\tilde{u}_{a^{\pm},M,\nu}$ be defined as in the beginning of Section 5. One can argue as in the proofs above and check that as $\delta,\epsilon\to 0$ a sequence of $\tilde{u}$’s converges uniformly to a solution $u^{*}$ of the transmission problem (5.14) with

[TABLE]

Since $\mathcal{F}^{*}$ is linear, the results Theorem 4.2 and Proposition 4.6 from Section 4 apply, and the rest of the arguments follow as in the proofs above.

∎

We remark that the proof of Theorem 1.2 above is easier than the one of Theorem 1.1. This is because the key results for the transmission problem in Section 4 are straightforward in the case of linear operators $\mathcal{F}$. In fact it can be shown that solutions to a linear transmission problem are of class $C^{\infty}(B_{1}^{\pm})$. It follows that the two-phase problem governed by (5.24) enjoys a quadratic improvement of flatness property and then Propositions 1.3 and 2.4 can be easily deduced.