Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources

1.   Introduction

In the last years the regularity theory for two phase problems governed by uniformly elliptic equations with distributed sources has reached a considerable level of completeness (see for instance the survey paper ^[10]) extending the results in the seminal papers ^[2,4] (for the Laplace operator) and in ^[17,18] (for concave fully non linear operators) to the inhomogeneous case, through a different approach first introduced in ^[7].

In particular the papers ^[15] and ^[8] provides optimal Lipschitz regularity for viscosity solutions and their free boundary for a large class of fully nonlinear equations.

Existence of a continuous viscosity solution through a Perron method has been established for linear operators in divergence form in ^[3] (homogeneous case) and in ^[9] (inhomogeneous case), and for a class of concave operators in ^[19]. The main aim of this paper is to adapt the Perron method to extend the results of ^[19] to the inhomogeneous case. Although we are largely inspired by the papers ^[3] and ^[9], the presence of a right hand side and the nonlinearity of the governing equation presents several delicate points, significantly in Section 6, which require new arguments.

We now introduce our class of free boundary problems and their weak (or viscosity) solutions.

Let ${\rm{Sym}}_n$ denote the space of $n \times n$ symmetric matrices and let $F:{\rm{Sym}}_n \rightarrow \mathbb R$ denote a positively homogeneous map of degree one, smooth except at the origin, concave and uniformly elliptic, i.e. such that there exist constants $0 < \lambda\le\Lambda$ with

where $\|M\| = \max_{|x| = 1}|Mx|$ denotes the $(L^2, L^2)$-norm of the matrix $M$.

Let $\Omega\subset{\mathbb{R}}^n$ be a bounded Lipschitz domain and $f_1, f_2 \in C(\Omega)\cap L^\infty(\Omega)$. We consider the following two-phase inhomogeneous free boundary problem (f.b.p. in the sequel).

Here $\nu = \nu(x)$ denotes the unit normal to the free boundary ${\mathcal{F}} = {\mathcal{F}}(u)$ at the point $x$, pointing toward $\Omega^+$, while the function $G(\beta, x, \nu)$ is Lipschitz continuous, strictly increasing in $\beta$, and

Moreover, $u_\nu^+$ and $u^-_\nu$ denote the normal derivatives in the inward direction to $\Omega^+(u)$ and $\Omega^-(u)$ respectively.

As we said, the main aim of this paper is to adapt Perron's method in order to prove the existence of a weak (viscosity) solution of the above f.b.p., with assigned Dirichlet boundary conditions

For any $u$ continuous in $\Omega$ we say that a point $x_0\in{\mathcal{F}}(u)$ is regular from the right (resp. left) if there exists a ball $B\subset\Omega^+(u)$ (resp. $B\subset\Omega^-(u)$) such that $\overline{B}\cap{\mathcal{F}}(u) = x_0$. In both cases, we denote with $\nu = \nu(x_0)$ the unit normal to $\partial B$ at $x_0$, pointing toward $\Omega^+(u)$.

Definition 1.1. A weak (or viscosity) solution of the free boundary problem (1.1) is a continuous function $u$ which satisfies the first two equality of (1.1) in viscosity sense (see Appendix A), and such that the free boundary condition is satisfied in the following viscosity sense:

(ⅰ) (supersolution condition) if $x_0\in \mathcal F$ is regular from the right with touching ball $B$, then, near $x_0$,

and

with equality along every non-tangential direction, and

(ⅱ) (subsolution condition) if $x_0\in \mathcal F$ is regular from the left with touching ball $B$, then, near $x_0$,

and

with equality along every non-tangential direction, and

We will construct our solution via Perron's method, by taking the infimum over the following class of admissible supersolutions ${\rm{class}}$.

Definition 1.2. A locally Lipschitz continuous function $w\in C(\overline{\Omega})$ is in the class ${\rm{class}}$ if

(a) $w$ is a solution in viscosity sense to

(b) if $x_0\in {\mathcal{F}}(w)$ is regular from the left, with touching ball $B$, then

and

with

(c) if $x_0\in {\mathcal{F}}(w)$ is not regular from the left then

The last ingredient we need is that of minorant subsolution.

Definition 1.3. A locally Lipschitz continuous function $\underline{u}\in C(\overline{\Omega})$ is a strict minorant if

(a) $\underline{u}$ is a solution in viscosity sense to

(b) every $x_0\in {\mathcal{F}}(\underline{u})$ is regular from the right, with touching ball $B$, and near $x_0$

where $\omega(r)\to0$ as $r\to0^+$, and

with

Our main result is the following.

Theorem 1.4. Let $g$ be a continuous function on $\partial\Omega$. If

(a) there exists a strict minorant $\underline{u}$ with $\underline{u} = g$ on $\partial \Omega$ and

(b) the set $\{w\in {\rm{class}} : w\geq \underline{u}, \ w = g\text{ on }\partial\Omega\}$ is not empty, then

is a weak solution of (1.1) such that $u = g$ on $\partial \Omega$.

Once existence of a solution is established, we turn to the analysis of the regularity of the free boundary.

Theorem 1.5. The free boundary ${\mathcal{F}}(u)$ has finite $(n-1)$-dimensional Hausdorff measure. More precisely, there exists a universal constant $r_{0}>0$ such that for every $r < r_{0},$ for every $x_{0}\in {\mathcal{F}}(u),$

Moreover, the reduced boundary ${\mathcal{F}}^{\ast }(u)$ of $\Omega ^{+}(u)$ has positive density in $\mathcal{H}^{n-1}$ measure at any point of $F(u)$, i.e. for $r < r_{0}$, $r_{0}$ universal

for every $x\in {\mathcal{F}}(u).$ In particular

Using the results in ^[8] we deduce the following regularity result.

Corollary 1.6. ${\mathcal{F}}(u)$ is a $C^{1, \gamma }$ surface in a neighborhood of ${\mathcal{H}}^{n-1}$ a.e. point $x_{0}\in {\mathcal{F}}(u).$

Notation. Constants $c$, $C$ and so on will be termed "universal" if they only depend on $\lambda$, $\Lambda$, $n$, $\Omega$, $\|f_i\|_\infty$ and $g$.

2.   Asymptotic developments

In this section we show that positive solutions of $F(D^2u) = f$ (with $f$ continuous up to the boundary) have asymptotically linear behavior at any boundary point which admits a touching ball, either from inside or from outside the domain. We need the following preliminary result.

Lemma 2.1. Let $r>0$, $\delta>0$, $\sigma>0$, $B_1^+: = B_1\cap\{x_1>0\}$ and let $E\subset \partial B_1^+\cap\{x_1>0\}$ be any subset such that there exists $\bar x\in E$ with

Let $u$ be the solution to

where $g_E$ is a cut-off function, $g_E = 1$ on $E$. If $r$ is sufficiently small then there exists a positive constant $C = C(\delta, \sigma)$ such that

Proof. We write

with ($F = F(M)$)

We have

Denote $u = v+w$ with $L_u v = 0$, $v = \delta\chi_E$ on $\partial B_1^+$ and $L_u w = r$, $w = 0$ on $\partial B_1^+$. By ^[11] we have that $v(e_1/2)\geq C\delta$, for some constant $C = C(n, \lambda, \Lambda, \sigma)$, and by the Boundary Harnack principle applied to $v$ and $u_1(x) = x_1$ we get that, in $B^+_{1/2}$, for some positive constants $c_0$ and $c_1$,

Put $z(x) = \frac{1}{2 \min a_{11}}(x_1-x_1^2)r$. The function $z$ is positive in $B^+_1$ and

Therefore $L_u(w+z)\leq 0$ in $B_1^+$ and $w+z\geq 0$ on $\partial B_1^+$. By the maximum principle $w\geq -c_2r x_1$ in $B_1^+$, where $c_2 = \frac{a_{11}}{\min a_{11}}>0$.

Summing up we get, in $B_{1/2}^+$,

for $r$ small enough, having $c_3>0$.

Lemma 2.2. Let $\Omega_1$ be a bounded domain with $0\in \partial \Omega_1$ and

Let $u$ be non-negative and Lipschitz in $\overline{\Omega}_1\cap B_2$, such that $F(D^2u) = f$ in ${\Omega}_1\cap B_2$ and that $u = 0$ in $\partial{\Omega}_1\cap B_2$ . Then there exists $\alpha\ge0$ such that

Proof. Let $\alpha_k = \sup\{\beta:u(x)\geq\beta x_1\text{ in }B^{+}_{1/k}\}$ for $k\ge 1$. Then the sequence $\{\alpha_k\}_k$ is increasing and $\alpha_k\leq L$ for any $k$, where $L$ is the Lipschitz constant of $u$. Let $\alpha = \lim_k \alpha_k$. By definition, $u(x)\geq \alpha x_1+o(|x|)$ in $B_1^+$, where $x = (x_1, x_2, ..., x_n)$.

Suppose by contradiction that $u(x)\neq \alpha x_1+o(|x|)$ in $B_1^+$. Then there exist a constant $\delta_0>0$ and a sequence $\{x_k\}_k = \{(x_{1, k}, x_{2, k}, ..., x_{n, k})\}_k\subset B_1^+$, with $|x_k| = r_k\rightarrow 0$, such that

Since $u$ is Lipschitz, a simple computation implies that

Let

The functions $u_k$ are defined in $B_1^+$ and, by assumption of homogeneity on $F$, we have

Moreover $u_k(x)\geq 0$ on $\partial B_1^+$ and $u_k\geq \delta_0/2$ in $E_k = \{x : x \in \partial B_1^+, x_1>0, |x-x_k|\leq\frac{\delta_0}{4L}\}$. We deduce that $u_k$ is a supersolution of (2.1). By comparison and Lemma 2.1, there exists $C>0$, not depending on $k$, such that

Writing $z = r_k x$ we obtain $u(z)\geq (\alpha_k +C)z_1$ in $B^+_{r_k/2}$. Choosing $k$, $k'$ in such a way that $\alpha_k + C > \alpha$ and $k'>2/r_k$ we obtain

a contradiction.

Lemma 2.3. Let $\Omega_1$ be a bounded domain such that, writing $B_1^-: = B_1\cap\{x_1 < 0\}$,

Let $u$ be non-negative and Lipschitz in $\overline{\Omega}_1\cap B_2(0)$, such that $F(D^2u) = f$ in ${\Omega}_1\cap B_2(0)$ and that $u = 0$ in $\partial{\Omega}_1\cap B_2(0)$ . Then there exists $\alpha\ge0$ such that

Proof. By assumption, we have that

Then we can extend $u$ as the zero function on $B_1^+\setminus \Omega_1$ so that it is a Lipschitz, non-negative solution to

Reasoning in a similar way as in Lemma 2.2, we define $\alpha_k = \inf\{\beta:u(x)\leq\beta x_1\text{ in }B_{1/k}\}$, $k\ge1$. Then $0\le \alpha_k < +\infty$ ($u$ is Lipschitz), and $\alpha_k\searrow \alpha\ge 0$, with $u(x)\leq \alpha x_1+o(|x|)$ in $B_1^+$. Again, let us suppose by contradiction that

where $\delta_0>0$ and $\{x_k\}_k = \{(x_{1, k}, x_{2, k}, ..., x_{n, k})\}_k\subset B_1^+$, is such that $|x_k| = r_k\rightarrow 0$. As before, such inequality propagates by Lipschitz continuity:

Defining the elliptic, homogeneous operator $F^*(M) = - F(-M)$, we have that the functions

solve

with $u_k(x)\geq 0$ on $\partial B_1^+$ and $u_k\geq \delta_0/2$ in $E_k = \{x : x \in \partial B_1^+, x_1>0, |x-x_k|\leq\frac{\delta_0}{4L}\}$. As a consequence, a contradiction can be obtained by reasoning as in Lemma 2.2.

Lemma 2.4. Let $\Omega_1$ be bounded domain with $0\in \partial \Omega_1$ and

Let $u$ be non-negative and Lipschitz in $\overline{\Omega}_1\cap B_2(0)$, such that $F(D^2u) = f$ in ${\Omega}_1\cap B_2(0)$ and that $u = 0$ in $\partial{\Omega}_1\cap B_2(0)$. Then there exists $\alpha\ge0$ such that

as $x\to0$ and either $x\in B_1(e_1)$ or $x\in\Omega$.

Proof. In both cases, we use the smooth change of variable

where $\psi(x')$ is smooth, with $\psi(x') = 1-\sqrt{1-|x'|^2}$ for $|x'|$ small. Then, by direct calculations, the function $\tilde u(y) = u(y_1 + \psi(y'), y')$ satisfies

where $\tilde F$ is still a uniformly elliptic operator. As a consequence the lemma follows by arguing as in the proofs of Lemmas 2.2, 2.3, with minor changes.

We conclude this section by providing a uniform estimate from below of the development coefficient $\alpha$, in case the touching ball is inside the domain.

Lemma 2.5. Let $u\in C(\overline{B_r(re_1)})$, $r\le1$, be such that

Moreover, assume that $u(re_1)\ge Cr$, for some $C>0$. Then

as $x\to0$, for $r\le \bar r$, where $c_1, c_2$ and $\bar r$ only depend on $\lambda, \Lambda, n$.

Proof. Let

Then

By Harnack's inequality [5,Theorem 4.3] we have that

where $c$ only depends on $\lambda, \Lambda, n$. We are in a position to apply Lemma A.2, which provides

as $x\to -e_1$, and the lemma follows.

Remark 2.6. Notice that the above results can be applied both to $F(D^2 u^+) = f_1$ in $\Omega^+(u)$ and to $F(D^2u^-) = f_2\chi_{\{u < 0\}}$ in $\Omega^-(w)$.

3.   The function $u^+$ is Lipschitz continuous

In this section we adapt the strategy developed in ^[3], in order to show that $u^+$ is locally Lipschitz. To this aim we need to use the following almost-monotonicity formula, provided in ^[6,14].

Proposition 3.1. Let $u_i$, $i = 1, 2$ be continuous, non-negative functions in $B_1$, satisfying $\Delta u_i \ge -1$, $u_1\cdot u_2 = 0$ in $B_1$. Then there exist universal constants $C_0$ and $r_0$, such that the functional

satisfies, for $0 < r\le r_0$,

Lemma 3.2. Let $w\in {\rm{class}}$. There exists $\tilde w\in {\rm{class}}$ such that

1. $F(D^2\tilde w) = f_1$ in $\Omega^+(\tilde w)$,

2. $\tilde w\leq w$, $\tilde w^- = w^-$, and

3. $\tilde w \geq \underline{u}$ in $\Omega$.

Proof. Let $w\in{\rm{class}}$ and $\Omega^+ = \Omega^+(w)$. We define

and

Since $\underline{u}^+\in{\mathcal{V}}$ we obtain that ${\mathcal{V}}$ is not empty and that $\underline{u} \le \tilde w \le w$. Moreover $\tilde w$ is a solution of the obstacle problem (see ^[13])

In particular, regularity results for the obstacle problem for fully nonlinear equations imply that $\tilde w$ is $C^{1, 1}$ in $\Omega^+$ (see ^[13]). To conclude that $\tilde w\in{\rm{class}}$, we need to show that the free boundary conditions in Definition 1.2 hold true. Let $x_0\in{\mathcal{F}}(\tilde w)$: if $x_0\in{\mathcal{F}}(w)$ too, then the free boundary condition follows from the fact that $\tilde w\le w$; otherwise, $x_0 \in \Omega^+$ is an interior zero of $\tilde w$, and the free boundary condition follows by the $C^{1, 1}$ regularity of $\tilde w$.

Lemma 3.3. Let $w\in {\rm{class}}$ with $F(D^2 w) = f_1$ in $\Omega^+(w)$, and let $x_0\in{\mathcal{F}}(w)$ be regular from the right. Then $u$ admits developments

with $0\le \alpha \le G(\beta, x_0, \nu(x_0))$, and

Proof. If $x_0$ is not regular from the left, then by definition of ${\rm{class}}$ the asymptotic developments hold with $\alpha = \beta = 0$ and there is nothing to prove. On the other hand, if $x_0$ is also regular from the left, then the asymptotic developments and the free boundary condition hold true by definition of ${\rm{class}}$ and by Lemma 2.4. Also in this case, if $\alpha = 0$ then there is nothing else to prove, thus we are left to deal with the case $\alpha>0$.

Reasoning as in [3,Lemma 3], see also [19,Lemma 4.3], one can show that

(recall that $\Phi(r)$ is defined in Proposition 3.1). On the other hand, since $F$ is concave,

The conclusion follows by combining Proposition 3.1 with (3.1).

Proposition 3.4. For every $D\subset\subset\Omega$ there exists a constant $L_D$, depending only on $D$, $G$, $\underline u$ and ${\rm{class}}$, such that

for every $x, y\in D$, $x\neq y$, and for every $w\in {\rm{class}}$ with $F(D^2 w) = f_1$ in $\Omega^+(w)$.

Proof. Let $x_0\in \Omega^+(w) \cap D$ such that

We will show that there exists $M>0$, not depending on $w$, such that

and the lemma will follow by Schauder estimates and Harnack inequality. By contradiction, let $M$ large to be fixed and let as assume that

Then Lemma 2.5 applies and we obtain

where $\alpha_M = c_1 M - c_2 r \|f\|_\infty > 0$ for $M$ sufficiently large. Then $x_0$ is regular from the right and Lemma 3.3 applies, with $\alpha_M \le \alpha \le G(\beta, x_0, \nu(x_0))$, providing

where $G^{-1}(\alpha): = \inf_{x, \nu}G^{-1}(\alpha, x, \nu)$. This provides a contradiction for $M$ sufficiently large.

Corollary 3.5. $u^+$ is locally Lipschitz and satisfies $F(D^2 u) = f_1$ in $\Omega^+(u)$.

4.   The function $u$ is Lipschitz continuous

Now we turn to the Lipschitz continuity of $u$.

Lemma 4.1. If $w_1, w_2\in {\rm{class}}$ then $\min\{w_1, w_2\}\in {\rm{class}}$.

Proof. This follows by standard arguments, see e.g. [9,Lemma 4.1].

To prove that $u$ is Lipschitz continuous we use the double replacement technique introduced in ^[9]. Let $w\in {\rm{class}}$ with $w(x_0) < 0$ and

Working on $\Omega_1$, we define

(which is non empty, for $R$ sufficiently small, as $\underline{u}^+\in{\mathcal{V}}_1$). Then

solves the obstacle problem (see ^[13])

On the other hand, working on $B$, let

(which is non empty, as $w^-\in{\mathcal{V}}_2$). Again,

solves the obstacle problem

Under the above notation, the double replacement $\tilde w$ of $w$, relative to $B$, is defined as

Lemma 4.2. Let $w\in {\rm{class}}$ with $w(x_0) = -h < 0$. There exists $\varepsilon >0$ (depending on ${\rm{dist}}(x_0, \partial\Omega)$ and $\underline{u}$) such that:

1. the double replacement $\tilde w$ of $w$, relative to $B_{\varepsilon h}(x_0)$, satisfies $\underline{u}\le \tilde w \leq w$ in $\Omega$;

2. $\tilde w < 0$ and $F(D^2\tilde w) = f_2$ in $B_{\varepsilon h}(x_0)$, with

3. $\tilde w \in {\rm{class}}$.

Proof. The inequality $w_1 \leq w$ in $\Omega_1$ follows by the maximum principle, while $w_2 \ge -w$ in $B$ because $w^-\in{\mathcal{V}}_2$. On the other hand, provided $\varepsilon$ is sufficiently small (depending on the Lipschitz constant of $\underline{u}$), we have that $\underline u < 0$ in $B: = B_{\varepsilon h}(x_0)$ and $u^*\in{\mathcal{V}}_1$, so that $w_1\ge \underline{u}$; finally, by the maximum principle in $\{w_2>0\}$, also $-w_2\ge \underline u$, and part 1. follows.

Turning to part 2., assume by contradiction that $\partial\{w_2>0\}\cap B_{\varepsilon h}(x_0)\neq\emptyset$. Then, by the regularity properties of the obstacle problem (4.2) (see ^[13]), we obtain that

Since $-w_2(x_0) \le w(x_0) = -h$, we obtain a contradiction for $\varepsilon$ sufficiently small. Then $w_2 > 0$ in $B_{\varepsilon h}(x_0)$, $w_2$ solves the equation by (4.2), and the remaining part of 2. follows by standard Schauder estimates and Harnack inequality.

Coming to part 3., the fact that $\tilde w$ satisfies (a) in Definition 1.2 follows by equations (4.1), (4.2) and by part 2. above, and we are left to check the free boundary conditions. For $\bar x\in{\mathcal{F}}(\tilde w)$, three possibilities may occur. If $\bar x\in {\mathcal{F}}(w)$ then, since $\tilde w \le w$, then $\tilde w$ has the correct asymptotic behavior both when $\bar x$ is regular and when it is not (recall that $G(0, \cdot, \cdot)>0$. If $\bar x\in\partial \{w_1>0\} \cap \Omega_1$, then we can use again the regularity of the obstacle problem]4.1 to obtain the correct asymptotic behavior. We are left to the final case, when $\bar x \in \partial B \cap \Omega^+(w)$. By Proposition 3.4, let us denote with $L$ the Lipschitz constant of $w$ in $B_{{\rm{dist}}(x_0, \partial\Omega)/2}(x_0)$. Then

Defining

we have that

Then Lemma A.1 applies, yielding

where

for universal $c_1$, $c_2$. Going back to $\tilde w$ we obtain

where $\nu = (\bar x-x_0)/|\bar x-x_0|$.

On the other hand, we can apply Lemma 2.5 to $(-w_2)_\varepsilon$, obtaining

where

for universal $c'_1$, $c'_2$, and thus

Comparing (4.3) and (4.4) we have that, choosing $\varepsilon$ small so that

the free boundary condition holds true.

Corollary 4.3. Let $u(x_0) = -h < 0$. There exist an non-increasing sequence $\{\tilde w_k\}\subset {\rm{class}}$, $\tilde w_k\ge \underline{u}$, and $\varepsilon>0$, depending on ${\rm{dist}}(x_0, \partial\Omega)$ and $\underline{u}$, such that the following hold:

1. $\tilde w_k(x_0) \searrow u(x_0)$;

2. $\tilde w_k < 0$ and $F(D^2\tilde w_k^-) = f_2$ in $B_{\varepsilon h}(x_0)$;

3. the sequence $\{\tilde w_k\}$ is uniformly Lipschitz in $B_{\varepsilon h/2}(x_0)$, with Lipschitz constant $L_0$ depending on ${\rm{dist}}(x_0, \partial\Omega)$.

4. $\tilde w_k \searrow u$ uniformly on $B_{h\varepsilon ilon/4}$

Proof. Let $u(x_0) = -h < 0$, $\{w_k\}\subset{\rm{class}}$ be such that $w_k\searrow u$ in some neighborhood of $x_0$ and $\{\tilde w _k\}\subset{\rm{class}}$ be the corresponding double replacements, as in Lemma 4.2. Then first three points are direct consequence of the lemma above, and we are left to prove that $\tilde w_k \searrow u$ uniformly on $B_{h\varepsilon ilon/4}$. By equicontinuity, $\tilde{w}_k \to \tilde{w}$ in $B_{\varepsilon ilon h/2}(x_0)$, and suppose by contradiction that $\tilde{w}(x_1)>u(x_1)$ for some $x_1\in B_{\varepsilon ilon h/4}(x_0)$. Then consider a new sequence $\{v_k\}_{k}$ converging to $u$ at $x_1$ and define $\{\tilde{u}_k\}_k$ as the double replacement of $\{\min\{\tilde{v}_k, \tilde{w}_k\} \}_{k}$ in $B_{\varepsilon ilon h/2}(x_0)$. Then $\tilde{u}_k\to\tilde{u}$, $\tilde{u}\leq\tilde{w}$ in $B_{\varepsilon ilon h /2}(x_0)$, $\tilde{u}(x_0) = \tilde{w}(x_0)$ and $\tilde{u}(x_1) < \tilde{w}(x_1)$. Since $F(D^2\tilde{w}) = F(D^2\tilde{u}) = f_2$ in $B_{\varepsilon ilon h/2}(x_0)$, this contradicts the strong maximum principle.

Corollary 4.4. For any $\overline{D}\subset\Omega$ there exists $\{w_k\}_{k}\subset {\rm{class}}$ such that $w_k\searrow u$ uniformly in $\overline{D}$. Furthermore, if $\overline{D}\subset \Omega^-(u)$, then each $w_k$ may be taken non-positive in $\overline{D}$.

Proof. The first part follows from the previous corollary. By compactness, it is enough to prove the second part for balls $\overline{B}_{\varepsilon}(x_0) \subset \Omega^-(u)$, with $\varepsilon$ small. Let $w_k \searrow u$ uniformly in $\overline B_{2\varepsilon}(x_0) \subset \Omega^-(u)$, and let

Let $\phi$ be such that

with $a$ and $\varepsilon$ positive and sufficiently small so that

(this is possible by explicit calculations, see for instance Lemma A.1); notice that this condition insure that $\phi$, extended to zero in $B_1$, is a supersolution in $B_2$ (when $c$ universal is suitably chosen). Since $u_\varepsilon \leq 0$ in $\overline B_2$, for $k$ sufficiently large $w_k\leq a/2$ in $\overline B_2$. Let us define

Then, by Lemma 4.1, the function

satisfies $\bar w_k \in {\rm{class}}$, $\bar w_k \leq 0$ in $\overline B_\varepsilon(x_0)$ and $\bar w_k \searrow u$ in $\overline B_{\varepsilon}(x_0)$, as required.

Corollary 4.5. $u$ is locally Lipschitz in $\Omega,$ continuous in $\overline \Omega$, $u = g$ on $\partial \Omega$. Moreover $u$ solves

5.   The function $u^+$ is non-degenerate

In this section we will show that $u^+$ is non-degenerate, in the sense of the following result.

Lemma 5.1. Let $x_0\in {\mathcal{F}}(u)$ and let $A$ be a connected component of $\Omega^+(u)\cap (B_{r}(x_0)\setminus \overline{B}_{r/2}(x_0))$ satisfying

for $r \leq r_0$ universal. Then

Moreover

where all the constants $C$ depend on $d(x, \partial \Omega)$ and on $\underline{u}.$

Corollary 5.2. ${\mathcal{F}}(w_k) \to {\mathcal{F}}(u)$ locally in Hausdorff distance and $\chi_{\{w_k>0\}}\to \chi_{\{u>0\}}$ in $L^1_{\mathit{\boldsymbol{loc}}}$.

The proof of the above result will follow by the two following lemmas.

Lemma 5.3. Let $u$ be a Lipschitz function in $\overline{\Omega}\cap B_1(0)$, with $0\in\partial \Omega$, satisfying

If there exists $c>0$ such that

then there exists a constant $C>0$ such that

for all $r \leq r_0$ universal.

Proof. Let $x_0 \in \Omega\cap B_1$, $\varepsilon = {\rm{dist}}(x_0, \partial\Omega)$, and let $L$ denote the Lipschitz constant of $u$. Then

We will show that, for $\delta>0$ to be fixed, there exists $x_1\in B_\varepsilon(x_0)$ such that

Then, iterating the procedure, one can conclude as in [9,Lemma 5.1].

Assume by contradiction that (5.2) does not hold. Then, defining the elliptic, homogeneous operator $F^*(M) = - F(-M)$, we infer that

Let $r(L) = 1-c/(4L)$; using the Harnack inequality we have that there exists $C(L)$ such that

provided both $\delta$ and $\varepsilon$ are sufficiently small (depending on $c$, $L$ and $\|f\|_\infty$). In terms of $u$, the previous inequality writes as

On the other hand, there exists $y_0\in\partial B_{r(L)\varepsilon}(x_0)$ such that ${\rm{dist}}(y_0, \partial\Omega) = (1-r(L))\varepsilon$ and hence

This is a contradiction, therefore (5.2) holds true.

Lemma 5.4. There exist universal constants $\bar r$, $\bar C$ such that

Proof. Let $x_0\in\{x\in\Omega^+(u):{\rm{dist}}(x, {\mathcal{F}}(u))\le \bar r\}$, with $\bar r$ universal to be specified later, and let $r: = {\rm{dist}}(x_0, {\mathcal{F}}(u))$. We distinguish two cases.

First let us assume that

In this case, for any $x\in{\mathcal{F}}(\underline u)$ we define

Notice that $\rho(x)>0$ for every $x$, since any point in ${\mathcal{F}}(\underline u)$ is regular from the right by assumption. Thus, recalling that $\underline u^+$ has linear growth bounded below by $\inf_{x, \nu}G(0, x, \nu)$, and noticing that $B_{3r/4}(x_0)\cap \Omega^+(\underline u)$ contains a ball of radius comparable with $r$ (at least for a suitable choice of $\bar r$):

where $\bar C$ only depends on $\underline u$.

On the other hand, in case

we have $\underline u\le 0$ in $B_{r/2}(x_0)$. By Corollary 4.4 we can find $\{w_k\}_k\subset{\rm{class}}$ converging uniformly to $u$ on some $D\supset B_{r}(x_0)$. By scaling

we need to find $\bar C$ universal such that $u_r(0)\ge \bar C$. Let us assume by contradiction that

Then by Harnack inequality

and, for $k$ sufficiently large,

Now, reasoning as in the proof of Corollary 4.4, let $\phi$ be such that

with $a$ and $r$ positive and sufficiently small so that $\nabla\phi(e_1/4) \cdot e_1 < \inf_{x, \nu}G(0, x, \nu)$, in such a way that $\phi$, extended to zero in $B_{1/4}$, is a supersolution in $B_{1/2}$. Then, choosing $\bar C < (a-r \|f_1\|_\infty)/C'$ we obtain that $w_k^r < \phi$ on $\partial B_{1/2}$ and then the functions

are continuous, while

satisfy $\bar w_k \in {\rm{class}}$, $\bar w_k \equiv 0$ in $\overline B_{r/4}(x_0)$. This is in contradiction with the fact that $u(x_0)>0$, and the lemma follows.

6.   The function $u$ is a supersolution

This section is devoted to the proof that $u$ satisfies the supersolution condition (ⅰ) in Definition 1.1. Thanks to Lemma 2.4 we only need to prove that, whenever $u$ admits asymptotic developments at $x_0\in{\mathcal{F}}(u)$, with coefficients $\alpha$ and $\beta$, then $\alpha\le G(\beta, x_0, \nu_{x_0})$. To do that, we need to distinguish the two cases $\beta>0$ and $\beta = 0$.

Lemma 6.1. Let $x_0\in{\mathcal{F}}(u)$, and

with

Then $\alpha\le G(\beta, x_0, \nu_{x_0})$.

Proof. Since $\beta >0$, then ${\mathcal{F}}\left(u\right)$ is tangent at $x_{0}$ to the hyperplane

in the following sense: for any point $x\in {\mathcal{F}}\left(u\right)$, dist$\left(x, {\mathcal{F}}\left(u\right) \right) = o\left(\left\vert x-x_{0}\right\vert \right).$ Otherwise we get a contradiction to the asymptotic development of $u$.

Let $\{w_k\}_k\subset {\rm{class}}$ be uniformly decreasing to $u$, as in Corollary 4.4. By the non-degeneracy of $u^+$ we have that, for $k$ large, $w_k$ can not remain strictly positive near $x_0$. Let $d_{k} = d_{H}\left({\mathcal{F}}\left(w_{k}\right), {\mathcal{F}}\left(u\right) \right)$ be the Hausdorff distance between the two free boundaries. In the ball $B_{2\sqrt{d_{k}}}\left(x_{0}\right)$, ${\mathcal{F}}\left(u\right)$ is contained in a strip parallel to $\pi$ of width $o\left(\sqrt{d_{k}} \right)$ and, since $d_{k}\rightarrow 0,$ ${\mathcal{F}}\left(w_{k}\right)$ is contained in a strip $S_{k}$ of width $d_{k}+o\left(\sqrt{d_{k}}\right) = o\left(\sqrt{d_{k}}\right)$.

Consider now the points $x_{k} = x_{0}-\sqrt{d_{k}}\nu$ and let $B_{k} = B_{r_{k}}\left(x_{k}\right)$ be the largest ball contained in $\Omega ^{-}\left(w_{k}\right)$ with touching point $z_{k}\in {\mathcal{F}}\left(w_{k}\right)$. Then $z_{k}\in S_{k}$ and, since $w_{k}\geq u$, from the asymptotic developments of $w_{k}$ and $u$ we have

since

Passing to the limit we infer

Reasoning in the same way on the other side th the points $y_{k} = x_{0}+\sqrt{ d_{k}}$ (and the same $z_{k},$ which are regular from the left), we get

From $\alpha _{k}\leq G\left(\beta _{k}, z_k, \nu_{k}\right)$, where $\nu _{k} = \left(x_{k}-z_{k}\right) /\left\vert x_{k}-z_{k}\right\vert$, we get $\alpha \leq G\left(\beta, x_0, \nu_{x_0} \right)$.

To treat the case $\beta = 0$ we need the following preliminary lemma.

Lemma 6.2. Let $v\geq 0$ continous in $B_1(x_0)$ be such that $\Delta v\geq -M$. Let

Then, for $r$ small,

Proof. We may assume $x_0 = 0$ and write $\Psi _{r}\left(0, v\right) = \Psi _{r}\left(v\right)$. Rescale setting $v_{r}\left(x\right) = v\left(rx\right) /r;$ we have $\Delta v_{r}\geq -rM$ and

Let $\eta \in C_{0}^{\infty }\left(B_{2}\right)$, $\eta = 1$ in $B_{1}$. Since $2\left\vert \nabla v_{r}\right\vert ^{2}\leq 2rMv_{r}+\Delta v_{r}^{2}$, we have:

so that

which is (6.1).

Lemma 6.3. Let $x_0\in{\mathcal{F}}(u)$, and

Then $\alpha\le G(\beta, x_0, \nu_{x_0})$.

Proof. As before, let $\{w_k\}_k\subset {\rm{class}}$ be uniformly decreasing to $u$, with $w_k$ that is not strictly positive near $x_0$, for $k$ large. The first part of the proof is exactly as in Lemma 6.3 of ^[9], until equation (6.2) below. For the reader's convenience, we recall such argument here.

For each $k$ we denote with

the largest ball centered at $x_{0}+\nu/{m}$ contained in $\Omega ^{+}(w_{k}),$ touching ${\mathcal{F}}(w_{k})$ at $x_{m, k}$ where $\nu _{m, k}$ is the unit inward normal of ${\mathcal{F}}(w_{k})$ at $x_{m, k}.$ Then up to proper subsequences we deduce that

and $B_{\lambda _{m}}(x_{0}+\nu/m)$ touches ${\mathcal{F}}(u)$ at $x_{m},$ with unit inward normal $\nu _{m}.$ From the behavior of $u^{+}$, we get that

and

Now since $w_{k}\in \mathcal{F}$, near $x_{m, k}$ in $B_{m, k}:$

and in $\Omega \setminus B_{m, k}$

with

(by Lemma 2.5 the touching occurs at a regular point, for $m, k$ large.) We know that

hence

and $\varepsilon ilon _{m}\rightarrow 0,$ as $m\rightarrow \infty.$ We have to show that

We assume by contradiction that $\bar\beta>0$. Acting as in [9,Lemma 6.3] we obtain, for $r$ small,

where

By concavity we have that $\Delta w_{k}^{\pm }\geq -M$ where $M = c\min \left(\| f_{1}\|_{\infty }, \|f_{2}\|_{\infty }\right)$. Lemma 6.2 implies

where $L$ is the uniform Lipschitz constant of $\{w_k^+\}_k$ (recall Lemma 3.4). Taking the $\liminf$ as $m, k\to\infty$ and using the uniform convergence of $w_{k}$ to $u$ we infer

Recalling that, by assumption, $u^-(x) = o(|x-x_0|)$ as $x\to x_0$, we have

and we get a contradiction.

7.   The function $u$ is a subsolution

In this section we want to show that $u$ is a subsolution according to Definition 1.1. Note that, if $x_{0}\in {\mathcal{F}}(u)$ is a regular point from the left with touching ball $B\subset \Omega ^{-}(u),$ then near to $x_{0}$

in $B,$ and

in $\Omega \backslash B.$ Indeed, even if $\beta = 0,$ then $\Omega ^{+}(u)$ and $\Omega ^{-}(u)$ are tangent to $\{\langle x-x_{0}, \nu \rangle = 0\}$ at $x_{0}$ since $u^{+}$ is non-degenerate. Thus $u$ has a full asymptotic development as in the next lemma. We want to show that $\alpha \geq G(\beta, x_{0}, \nu).$ We follow closely ^[3] and ^[9].

Lemma 7.1. Assume that near $x_{0}\in {\mathcal{F}}(u)$,

with $\alpha >0$, $\beta \geq 0$. Then

Proof. Assume by contradiction that $\alpha < G(\beta, x_{0}, \nu)$. We construct a supersolution $w\in {\rm{class}}$ which is strictly smaller than $u$ at some point, contradicting the minimality of $u$. Let $u_{0}$ be the two-plane solution, i.e.

Suppose that $\alpha \leq G(\beta, x_{0}, \nu)-\delta _{0}$ with $\delta _{0}>0.$ Fix $\zeta = \zeta (\delta _{0})$, to be chosen later. By Corollary 4.4, we can find $w_{k}\in F\searrow u$ locally uniformly and, for $r$ small, $k$ large, the rescaling $w_{k, r}$ satisfies the following conditions:

if $\beta >0,$ then

if $\beta = 0,$ then

and

In particular,

If $\beta >0$, let $v$ satisfy

where $\phi \geq 0$ is a cut-off function, $\phi \equiv 0$ outside $B_{1/2},$ $\phi \equiv 1$ inside $B_{1/4}.$

For $\beta = 0,$ replace the second equation with $v = 0.$

Along the new free boundary, ${\mathcal{F}}(v) = \{\langle x, \nu \rangle = -\zeta +\varepsilon ilon \phi (x)\}$ we have the following estimates:

with $c, C$ universal.

Indeed,

is a solution of

Thus, by standard $C^{1, \gamma }$ regularity estimates (see [16,Theorem 1.1])

which gives the desired bound. Similarly, one gets the bound for $v_{\nu }^{-}.$

Hence, since $\alpha \leq G(\beta, x_{0}, \nu (x_{0}))-\delta _{0}$, say for $\varepsilon = 2\zeta$ and $\zeta, r$ small depending on $\delta _{0}$

and the function,

is still in ${\rm{class}}.$ However, the set

contains a neighborhood of the origin, hence rescaling back $x_{0}\in \Omega ^{-}(\bar{w}_{k})$. We get a contradiction since $x_{0}\in F(u)$ and $\Omega ^{+}(u)\subseteq \Omega ^{+}(\bar{w}_{k}).$

8.   Properties of the free boundary

In this section we prove the weak regularity properties of the free boundary. Both statements and proofs are by now rather standard and follows the papers ^[3] and ^[9] for problems governed by homogeneous and inhomogeneous divergence equations, respectively. Thus we limit ourselves to the few points in which differences from the previous cases emerge. Denote by $\mathcal{N}_{\varepsilon }\left(A\right)$ an $\varepsilon$-neighborhood of the set $A.$ The following lemma provides a control of the ${\mathcal{H}}^{n-1}$ measure of ${\mathcal{F}}\left(u\right)$ and implies that $\Omega ^{+}\left(u\right)$ is a set of finite perimeter.

Lemma 8.1. Let $u$ be our Perron solution. Let $x_{0}\in {\mathcal{F}}\left(u\right) \cap B_{1}.$ There exists a positive universal $\delta_{0} < 1$ such that, for every $0 < \varepsilon < \delta \leq \delta _{0}$, the following quantities are comparable:

1. $\frac{1}{\varepsilon }\left\vert \left\{ 0 < u < \varepsilon\right\} \cap B_{\delta }\left(x_{0}\right) \right\vert$,

2. $\frac{1}{\varepsilon }\left\vert \mathcal{N}_{\varepsilon }\left({\mathcal{F}}\left(u\right) \right) \cap B_{\delta }\left(x_{0}\right) \right\vert$,

3. $N\varepsilon ^{n-1}$, where $N$ is the number of any family of balls of radius $\varepsilon,$ with finite overlapping, covering ${\mathcal{F}}\left(u\right) \cap B_{\delta }\left(x_{0}\right)$,

4. ${\mathcal{H}}^{n-1}\left({\mathcal{F}}\left(u\right) \cap B_{\delta }\left(x_{0}\right) \right).$

Proof. From ^[3], it is sufficient to prove the following two equivalences:

and

with universal constants $c_{1}, c_{2}, C_{1}, C_{2}.$

Since $F \left(D^{2}u\right) = \inf_{\alpha }L_{\alpha }u$ where $L_{\alpha }$ is a uniformly elliptic operator with constant coefficients and ellipticity constant $\lambda, \Lambda$, we have $L_{\alpha }u^{+}\geq f_{1}$ in $\Omega ^{+}\left(u\right)$. Fix $\alpha = \alpha _{0}$ and set

The upper bound in (8.1) follows by the Lipschitz continuity of $u$. The lower bound follows from $\sup_{B_{\varepsilon }\left(x_{0}\right) }u^{+}\geq c\varepsilon$, $c$ universal, $\inf_{B_{\varepsilon }\left(x_{0}\right) }u^{+} = 0$, the Lipschitz continuity of $u$, and the Poincaré inequality (see [1,Lemma 1.15]).

To prove (8.2), rescale by setting

Then $Lu_{\delta }\geq \delta$ $f_{1}^{\delta }$ in $\Omega ^{+}\left(u^{\delta }\right) \cap B_{1}.$ For $0 < \varepsilon < \delta$, let

We have:

since $\nabla u_{\varepsilon, s} = \nabla u_{\delta }\cdot \chi _{\{s/\delta < u_{\delta } < \varepsilon /\delta \}}.$

By uniform ellipticity, since $u^{+}$ is Lipschitz and $f_{1}$ is bounded, we get ($\delta < 1$)

with $C$ universal. Letting $s\rightarrow 0$ and rescaling back, we obtain the upper bound in (8.2).

For the lower bound, let $V$ be the solution to

with $\sigma$ to be chosen later. By standard estimates, see for example ^[12], $V\leq C\sigma ^{2-n}$ and $-\langle A\nabla V, \nu \rangle \sim C^{\ast }$ on $\partial B_{1},$ with $C^{\ast }$ independent of $\sigma.$ By Green's formula

since $V = 0$ on $\partial B_{1}.$ We estimate

since $u$ is Lipschitz and $0\leq u_{\varepsilon, 0}\leq \varepsilon /\delta$. From (8.4) and (8.5) and the fact that $\langle A_{\delta }\nabla V, \nu \rangle \sim -C^{\ast }$ on $\partial B_{1}$ we deduce that

Thus using that $u^{+}$ is non-degenerate and choosing $\sigma$ small enough (universal) we get that ($\delta >\varepsilon$)

On the other hand in $\{0 < u_{\delta }^{+} < \varepsilon /\delta \}\cap B_{1},$

Combining (8.6), (8.7) and using the ellipticity of $A$ we get that

From the estimate on $V$ we obtain that for $\delta$ small enough

for some $C$ universal. Rescaling, we obtain the desired lower bound.

Lemma 8.1 implies that $\Omega ^{+}(u)\cap B_{r}(x)$, $x\in {\mathcal{F}}(u)$, is a set of finite perimeter. Next we show that in fact this perimeter is of order $r^{n-1}.$

Theorem 8.2. Let $u$ be our Perron solution. Then, the reduced boundary of $\Omega ^{+}(u)$ has positive density in $\mathcal{H}^{n-1}$-measure at any point of ${\mathcal{F}}(u)$, i.e. for $r < r_{0}$, $r_{0}$ universal,

for every $x\in {\mathcal{F}}(u).$

Proof. The proof follows the lines of Corollary 4 in ^[3] and Theorem 8.2 in ^[9]. Let $w_{k}\in {\rm{class}}$, $w_{k}\searrow u$ in $\overline{ B}_{1}$ and $L$ as in Lemma 8.1. Then $\Omega ^{+}\left(u\right) \subset \subset \Omega ^{+}\left(w_{k}\right)$ and $Lw_{k}\geq F\left(D^{2}w_{k}\right) = f_{1}$ in $\Omega ^{+}(u).$ Let $x_{0}\in {\mathcal{F}}(u).$ We rescale by setting

Let $V$ be the solution to (8.3). Since $\nabla w_{k, r}$ is a continuous vector field in $\overline{\Omega _{r}^{+}(u_{r})\cap B_{1}},$ we can use it to test for perimeter. We get

Using the estimates for $V$ and the fact that the $w_{k}$ are uniformly Lipschitz, we get that

As in ^[3] we have, as $k\rightarrow \infty$,

and

Passing to the limit in (8.8) and using all of the above we get

Since $u$ is Lipschitz and non-degenerate, for $\sigma$ small

and using the estimate for $\langle A\nabla V, \nu \rangle$

Also, since $f_{1}^{r}$ is bounded,

Hence choosing first $\sigma$ and then $r$ sufficiently small we get that

$\tilde{C}$ universal.

A.   Some explicit barrier functions

For the reader's convenience we collect here some explicit barrier functions which arise frequently in our arguments. Their proof is based on comparison arguments, together with the well known chain of inequalities

where $\mathcal{P}^-_{\lambda/n, \Lambda}$ denotes the lower Pucci operator, and $c = c(\lambda, \Lambda, n)> 0$ since $F$ is concave (see ^[5] for further details).

Lemma A.1 (Barrier for subsolutions). Let $u$ satisfy

Then

as $x\to e_1$, where the positive constants $c_1$, $c_2$ only depend on $\lambda, \Lambda, n$.

Proof. By comparison and (A.1) we infer that $u\ge \phi$ in $B_{2} \setminus \overline{B}_{1}$, where $\phi$ solves

for a universal $c$. Then direct calculations show that, for $n\ge3$,

where

Then the Lemma follows by choosing

The proof in dimension $n = 2$ is analogous.

Lemma A.2 (Barrier for supersolutions). Let $u$ satisfy

Then

as $x\to -2e_1$, whenever $r\le \bar r$, where the positive constants $c_1$, $c_2$ and $\bar r$ only depend on $\lambda, \Lambda, n$.

Proof. By comparison and (A.1) we infer that $u\ge \phi$ in $B_{2} \setminus \overline{B}_{1}$, where $\phi$ solves

Then direct calculations show that

and

To check this, one needs to choose $r\le\bar r = \bar r(\gamma)$, in such a way that $D^2\phi(x)$ has exactly one positive eigenvalue, for $1\le|x|\le2$. Then the Lemma follows by choosing

Acknowledgments

Work partially supported by the INDAM-GNAMPA group. G. Verzini is partially supported by the project ERC Advanced Grant 2013 n. 339958: "Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT"and by the PRIN-2015KB9WPT Grant: "Variational methods, with applications to problems in mathematical physics and geometry".

Conflict of interest

The authors declare no conflict of interest.