Structure of singularities in the nonlinear nerve conduction problem

TL;DR

This paper characterizes the structure of singular points on the free boundary in a nonlinear nerve conduction model, revealing geometric properties and stratification near these points, especially in two dimensions.

Contribution

It provides a detailed analysis of the free boundary's singularities in a nonlinear PDE model of nerve impulses, including geometric descriptions and stratification methods.

Findings

Near flat singular points, the free boundary consists of four $C^1$ arcs in 2D.

In the linear case, the singular set decomposes into degenerate and flat points.

Blow-ups of solutions at certain points are homogeneous functions.

Abstract

We give a characterisation of the singular points of the free boundary $\partial \{u>0\}$ for viscosity solutions of the nonlinear equation \begin{equation}F(D^2 u)=-\chi_{\{u>0\}},\tag{0.1} \end{equation} where $F$ is a fully nonlinear elliptic operator and $\chi$ the characteristic function. The equation (0.1) models the propagation of a nerve impulse along an axon. We analyse the structure of the free boundary $\partial\{ u>0\}$ near the singular points where $u$ and $\nabla u$ vanish simultaneously. Our method uses the stratification approach developed in [DK18]. In particular, when $n=2$ we show that near a rank-2 flat singular free boundary point $\partial\{ u>0\}$ is a union of four $C^1$ arcs tangential to a pair of crossing lines. Moreover, if $F$ is linear then the singular set of $\partial\{ u>0\}$ is the union of degenerate and rank-2 flat points. We also provide an application of the boundary Harnack principles to study the higher order flat degenerate points and show that if $\{u<0\}$ is a cone then the blow-ups of $u$ are homogeneous functions.

Equations

$$F(D^{2}u)=-{\chi }_{\{u>0\}},$$

$$F(D^{2}u)=-{\chi }_{\{u>0\}}\text{mbox}in\Omega ,$$

$$\left\{\begin{array}[]{lll}u_{t}=r(x)\Delta u+F(u,\vec{v}),\\ \vec{v}_{t}=G(u,\vec{v}),\end{array}\right.$$

$$\Delta u=-{\chi }_{\{u>0\}}$$

$$\lambda \Vert N\Vert \le F(M+N)-F(M)\le \Lambda \Vert N\Vert ,M\in \mathcal{S},$$

$$\lambda {\left|\xi \right|}^2\le F_{ij}\left(M\right){\xi }_i{\xi }_j\le \Lambda {\left|\xi \right|}^2,$$

$${\mathcal{M}}^{-}\left(M,\lambda ,\Lambda \right)=\lambda \sum _{e_i>0}{e}_i+\Lambda \sum _{e_i<0}{e}_i,{\mathcal{M}}^{+}\left(M,\lambda ,\Lambda \right)=\Lambda \sum _{e_i>0}{e}_i+\lambda \sum _{e_i<0}{e}_i,$$

$$\Vert u{\Vert }_{C^{2,\alpha }(B_{\frac{1}{2}})}\le C\left(\Vert u{\Vert }_{L^{\infty }(B_1)}+|F(0)|\right),$$

$$\Vert u{\Vert }_{C^{1,1}(B_{1/11})}\le C\Vert u{\Vert }_{L^{\infty }(B_1)},$$

$$v<w\text{in}D.$$

$$C^{-1}\frac{u(A)-\sigma v(A)}{v(A)}\le \frac{u-\sigma v}{v}\le C\frac{u(A)-\sigma v(A)}{v(A)}.$$

$${\partial }_{2}w\ge 0\text{mbox}in{D}_{\delta }=\{|x_1|\le \delta ,f(x_1)<x_2\le M\delta \}.$$

$$C_{R}w(Rx)=w(x),R\in (0,1),C_R=\frac{1}{{\sup }_{{\mathcal{K}}_R}w}.$$

$$w(x)=|x{|}^{\kappa }w\left(\frac{x}{|x|}\right).$$

$$\left\{\begin{array}[]{lll}F(D^{2}u)=-\chi_{\{u>0\}}&\mbox{in}\ D,\\ u=g&\mbox{on}\ \partial D,\end{array}\right.$$

$$\left\{\begin{array}[]{lll}\beta_{\varepsilon}(t)\geq\chi_{\{t>0\}}&\mbox{on}\ \mathbb{R},\\ \beta_{\varepsilon^{\prime}}(t)\leq\beta_{\varepsilon}(t)&\mbox{if}\ \varepsilon^{\prime}<\varepsilon,\\ \lim_{\varepsilon\to 0}\beta_{\varepsilon}(t)=\chi_{\{t>0\}}&t\in\mathbb{R}.\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}F(D^{2}v)=-\beta_{\varepsilon}(v)&\mbox{in}\ D,\\ v=g&\mbox{on}\ \partial D.\end{array}\right.$$

$$F(D^2{u}_{{\epsilon }^{\prime }})=-{\beta }_{{\epsilon }^{\prime }}(u_{{\epsilon }^{\prime }})\ge -{\beta }_{\epsilon }(u_{{\epsilon }^{\prime }}).$$

$$v\le u_{\epsilon },u_{{\epsilon }^{\prime }}\le u_{\epsilon }.$$

$$\underset{B_r(x_0)}{\inf }u>-c_{n,\gamma }{r}^2$$

$$b(x)=\left\{\begin{array}[]{lll}C(1-|x|^{2})&\mbox{if}\ |x|\leq 1,\\ \phi(x)-\phi(1)&\mbox{if}\ |x|>1,\end{array}\right.$$

$$\phi(x)=\left\{\begin{array}[]{lll}-\log|x|&\mbox{if}\ n=2,\\ \frac{1}{\gamma}{|x|^{-\gamma}}&\mbox{if}\ n\geq 3,\end{array}\right.$$

$$F(D^{2}b)=\left\{\begin{array}[]{lll}-2CF(\delta_{ij})&\mbox{if}\ |x|\leq 1,\\ -\frac{1}{|x|^{\gamma+2}}F(\delta_{ij}-(\gamma+2)\frac{x_{i}x_{j}}{|x|^{2}})&\mbox{if}\ |x|>1.\end{array}\right.$$

$$F({\delta }_{ij}-(\gamma +2)\frac{x_i{x}_j}{|x{|}^2})\le {\mathcal{M}}^{+}({\delta }_{ij}-(\gamma +2)\frac{x_i{x}_j}{|x{|}^2})=0,|x|>1.$$

$$F(D^{2}b)\ge -\frac{1}{|x{|}^{\gamma +2}}{\mathcal{M}}^{+}({\delta }_{ij}-(\gamma +2)\frac{x_i{x}_j}{|x{|}^2})=0,|x|>1.$$

$$\frac{1}{{\rho }^2}\hat{b}(\rho x)=\frac{1}{{\rho }^{\gamma +2}\gamma }\left[\frac{1}{|x{|}^{\gamma }}-{\rho }^{\gamma }\right],$$

$$\frac{\hat{b}(r)}{{\rho }^2}$$

$$P_2:=\left\{p(x)=\sum a_{ij}{x}_i{x}_j,\text{mbox}foranyx\in {\mathbb{R}}^n,\text{mbox}with\Vert p{\Vert }_{L^{\infty }(B_1)}=1\right\},$$

$$S(p,x_0):=\{x\in {\mathbb{R}}^n:p_{x_0}(x)=0\}.$$

$${\rm{HD}}\Big{(}\partial\{u>0\}\cap B_{r}(x_{0}),S(p,{x_{0}})\cap B_{r}(x_{0})\Big{)}<\delta\,r.$$

Full text

[TABLE]

where $F$ is a fully nonlinear elliptic operator and $\chi$ the characteristic function. The equation (0.1) models the propagation of a nerve impulse along an axon.

We analyze the structure of the free boundary $\partial\{u>0\}$ near the singular points where $u$ and $\nabla u$ vanish simultaneously. Our method uses the stratification approach developed in [DK].

In particular, when $n=2$ we show that near a rank-2 flat singular free boundary point $\partial\{u>0\}$ is a union of four $C^{1}$ arcs tangential to a pair of crossing lines. Moreover, if $F$ is linear then the singular set of $\partial\{u>0\}$ is the union of degenerate and rank-2 flat points.

We also provide an application of the boundary Harnack principles to study the higher order flat degenerate points, and show that if $\{u<0\}$ is a cone then the blow-ups of $u$ are homogeneous functions.

1. Introduction

In this paper we study the free boundary problem

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$ is a given bounded domain with $C^{2,\alpha}$ boundary, $\chi_{\{u>0\}}$ the characteristic function function of $\{u>0\}$, and $F$ a convex fully nonlinear elliptic operator satisfying some structural conditions. The partial differential equation (1.1) appears in a model of the nerve impulse propagation [Feroe], [Pauwelussen], [Rinzel].

It comes from the following linearized diffusion system of FitzHugh

[TABLE]

where $u(x,t)$ is the voltage across the nerve membrane at distance $x$ and time $t$, and components of $\vec{v}=(v^{1},\dots,v^{k})$ model the conductance of the membrane to various ions [Feroe]. Mckean suggested to consider $F(u,\vec{v})=-u+\chi_{\{u>0\}}$ [McKean]. Due to the homogeneity of the equation the linear term disappears after quadratic scaling, so we neglect it.

The linearized steady state equation

[TABLE]

also arises in a solid combustion model [Monneau-W], and the composite membrane problem, see [Kenig], also [CK] for its variational formulation.

A chief difficulty is to analyze the free boundary near singular points where both $u$ and $\nabla u$ vanish. The main technique used in [Monneau-W], [Kenig], [CK] is a monotonicity formula, which is not available for the nonlinear equations. The aim of this paper is to use the boundary Harnack principles and anisotropic scalings to develop a new approach to circumvent the lack of the monotonicity formula and obtain some of the main results from [Monneau-W] and [Lee-Sh] for the fully nonlinear case. More precisely, in this paper we address the optimal regularity, degeneracy and the shape of the free boundary near the singular points.

One of the main results in [Monneau-W] concerns the cross shaped singularities in $\mathbb{R}^{2}$. It follows from the classification of homogeneous solutions and an application of the monotonicity formula introduced in [Monneau-W]. For nonlinear equations this method cannot be applied. We remark that the degenerate case (i.e. when $u(x)=o(|x-x_{0}|^{2})$ near a free boundary point $x_{0}$) cannot be treated by the monotonicity formula introduced in [Monneau-W] because it does not provide any qualitative information about $u$, see Proposition 5.1 [Monneau-W]. Another approach was recently used in [ST].

It is well known that the strong solutions of (1.3) may not be $C^{1,1}_{loc}$, see [Chemin] Proposition 5.3.1. However, if $F=\Delta$ then $\nabla u$ is always log-Lipschitz continuous [Yudovich] Lemma 2.1. For general elliptic operators one can show that $\nabla u$ is $C^{\alpha}$ for every $\alpha\in(0,1)$, [CC-Fully], see also Remark 2.2 in Section 2.

Another approach, based on Harnack inequalities, had been developed by Tolksdorf to prove the existence of homogeneous solutions for $\Delta_{p}u=0$ in cones [Tolksdorf] . We employ this approach in Section 6.

The problem (1.3) has some resemblance with the classical obstacle problem [Caff-cpde]. For the fully nonlinear operators the obstacle problem has been studied in [Lee] for one phase and in [Xavier] for the thin obstacle.

The paper is organized as follows: In Section 2 we state some technical results. In Section 3 we prove the existence of viscosity solutions using a penalization argument. We also show the existence of a maximal solution and establish its non-degeneracy. Section 4 contains the proof of the following dichotomy: either the free boundary points are rank-2 flat or the solution has quadratic growth. As a consequence we show that if $n=2$ then near a rank-2 flat point the free boundary is a union of four $C^{1}$ curves tangential to a pair of crossing lines. This is done in Section 5. Finally, in Section 6 we use a boundary Harnack principle to prove the homogeneity of blow-ups near conical free boundary points.

2. Technical results

Throughout this paper $B_{r}(x)$ denotes the open ball of radius $r$ centered at $x\in\mathbb{R}^{n}$ and $B_{r}=B_{r}(0)$. For continuous function $u$ we let $u=u^{+}-u^{-}$, $u^{+}=\max(0,u)$, $\Omega^{+}(u)=\{u>0\},\Omega^{-}(u)=\{u<0\}$, and $\partial_{\rm{sing}}\{u>0\}$ is the singular subset of the free boundary $\partial\{u>0\}$, where $u=|\nabla u|=0$.

We shall make two standing assumptions on the operators under consideration. To formulate them we let $\mathcal{S}$ be the space of $n\times n$ symmetric matrices and $\mathcal{S}^{+}({\lambda,\Lambda})$ positive definite symmetric matrices with eigenvalues bounded between two positive constants $\lambda$ and $\Lambda$.

• $\bf F1^{\circ}$

The operator $F:\mathcal{S}\subset\mathbb{R}^{n\times n}\to\mathbb{R}$ is uniformly elliptic, i.e. there are two positive constants $\lambda,\Lambda$ such that

[TABLE]

for every nonnegative matrix $N$.

• $\bf F2^{\circ}$

$F$ is smooth except the origin and homogeneous of degree one $F(tM)=tF(M),t\in\mathbb{R}$, and $F(0)=0$.

For smooth $F$ the hypothesis $\bf F1^{\circ}$ is equivalent to

[TABLE]

where $F_{ij}(S)=\frac{\partial F(S)}{\partial s_{ij}},S=[s_{ij}]$.

Typically, $F(M)=\sup_{t\in\mathcal{I}}A_{ij,t}M_{ij}$, where $\mathcal{I}$ is the index set and $A_{ij,t}\in\mathcal{S}^{+}({\lambda,\Lambda})$ such that $\lambda\left|\xi\right|^{2}\leq A_{ij,t}\xi_{i}\xi_{j}\leq\Lambda\left|\xi\right|^{2}$. Notice that if $w_{t}(x)=w(A_{t}^{\frac{1}{2}}x)$ then $\Delta w_{t}=A_{ij,t}w_{ij}$.

We also define Pucci’s extremal operators

[TABLE]

where $e_{1}\leq e_{2}\leq\dots\leq e_{n}$ are the eigenvalues of $M\in\mathcal{S}$.

Definition 2.1**.**

A continuous function $u$ is said to be a viscosity solution of $F(D^{2}u)=-\chi_{\{u>0\}}$, if the equation $F(D^{2}v(x_{0}))=-\chi_{\{u>0\}}$ holds pointwise, whenever at $(x_{0},u(x_{0}))$ the graph of $u$ can be touched from above and below by paraboloids $v$.

Remark 2.2**.**

If $F$ is concave and $u$ is a viscosity solution of $F(D^{2}u)=0$ in $B_{1}$ then

[TABLE]

where $0<\alpha<1$ and $C$ are universal constants, see Theorem 6.6 [CC-Fully]. If $F$ is convex or concave then for the viscosity solutions of $F(D^{2}u)=0$ we still have the estimate

[TABLE]

(see (6.14) and Remark 1 on page 60 in [CC-Fully]). Theorem 5.2 is the only place where we require $F$ to be convex.

Under assumptions $\bf F1^{\circ}-F2^{\circ}$ the classical weak and strong comparison principles are valid for the viscosity solutions [CC-Fully]. Moreover, we have the strong and Hopf’s comparison principles.

Lemma 2.3** (Strong comparison principle).**

Suppose $v\in C^{2}(D),w\in C^{1}(D)$, and $\nabla v\not\equiv 0$ in a bounded domain $D$. Let $F(D^{2}v)\geq 0\geq F(D^{2}w)$ in $D\subset\mathbb{R}^{n}$ in viscosity sense, $v\leq w$, and $v,w$ are not identical, then

[TABLE]

See Theorem 3.1 [Giga].

Lemma 2.4** (Hopf’s comparison principle).**

Let $B$ be a ball contained in $D$ and assume that $w\in C^{1}(D),v\in C^{2}(D)$ and that $\nabla v\not=0$, in $B$. Let $v$ and $w$ be a viscosity subsolution and a supersolution of $F(D^{2}u)=0$, respectively. Moreover, suppose that $v<w$, in $B$, and that $v(x_{0})=w(x_{0})$, for some $x_{0}\in\partial B$. Then, $\nabla v(x_{0})\not=\nabla w(x_{0})$.

See Theorem 4.1 [Giga].

One of the main tools in our analysis is the boundary Harnack principle. As before, we assume that $F$ is smooth, homogeneous of degree 1, uniformly elliptic with ellipticity constants $\lambda$ and $\Lambda$, and $F(0)=0$. We use the following notation: $f(x^{\prime}),x^{\prime}\in B_{1}^{\prime}\subset\mathbb{R}^{n-1}$ is Lipschitz continuous function with Lipschitz constant $M>1,f(0)=0$, $\Omega_{r}=B_{r}^{\prime}\times[-rM,rM]\cap\{x_{n}>f(x^{\prime})\}$, $\Delta_{r}=B_{r}^{\prime}\times[-rM,rM]\cap\{x_{n}=f(x^{\prime})\}$, $A=e_{n}M/2$.

Then we have the following Harnack principle, see [Wang].

Theorem 2.5**.**

Assume $\bf F1^{\circ}-F2^{\circ}$ hold and $F$ is either concave or convex. Let $u,v$ be two nonnegative solution of $F(D^{2}u)=0$ in $\Omega_{1}$ that equal 0 along $\Omega_{1}\setminus S$. Suppose also that $u-\sigma v\geq 0$ in $\Omega_{1}$ for some $\sigma\geq 0$. Then for some constant $C$, depending only on $\lambda,\Lambda,n$ and the Lipschitz character of $\Omega_{1}$, we have in $\Omega_{\frac{1}{2}}$

[TABLE]

Furthermore, as in [C-1] (see also [Wang] Section 2) one can show that the nonnegative solutions in $\Omega_{1}$ are monotone in $\Omega_{\delta_{0}}$ for some universal $\delta_{0}$. We state this only in two spatial dimensions.

Theorem 2.6**.**

Let $w$ be a viscosity solution of $F(D^{2}w)=0,w\geq 0$ in $D=\{|x_{1}|\leq 1,f(x_{1})<x_{2}\leq M\}$, $M=\|f\|_{C^{0,1}}$. Assume $\bf F1^{\circ}-F2^{\circ}$ hold and $F$ is either concave or convex. Then there is $\delta=\delta(M)$ such that

[TABLE]

In [Wang] Theorem 2.6 is stated for concave operator $F$, however the concavity is needed only to assure that locally the viscosity solutions of the homogeneous equation are locally $C^{2,\alpha}$ regular, see Remark 1.2 in [Wang]. Since in the proofs of Lemmata 2.1-2.5 in [Wang] one needs only $C^{1,\alpha}$ regularity of the solutions then in view of Remark 2.2 we see that Theorem 2.6 continues to hold for convex $F$, see [Feldman].

Finally, we give a characterization of homogeneity, see Theorem 2.1.1 [Tolksdorf] for a proof.

Lemma 2.7**.**

Let $w\in L^{\infty}(\mathcal{K}_{1})\setminus\{0\}$ where $\mathcal{K}_{1}=\mathcal{K}(0,1)$ such that

[TABLE]

Then there is a $\kappa\in[0,\infty)$ such that

[TABLE]

3. Existence and nongeneracy

In this section we prove the existence of viscosity solutions and the non-degeneracy of maximal solutions.

3.1. Existence of viscosity solutions

Definition 3.1**.**

A continuous function $u$ is said to be a viscosity subsolution of $F(D^{2}u)=-\chi_{\{u>0\}}$, if the inequality $F(D^{2}v(x_{0}))\geq-\chi_{\{u>0\}}$ holds pointwise, whenever at $(x_{0},u(x_{0}))$ the graph of $u$ can be touched from below by a paraboloid $v$. Moreover, $u$ is said to be a strict subsolution if the inequality above is strict.

Definition 3.2**.**

A viscosity solution $u$ of $F(D^{2}u)=-\chi_{\{u>0\}}$ is said to be maximal in $D$ if for every strong subsolution $v$ satisfying $v\leq u$ on $\partial D^{\prime}$ for some subdomain $D^{\prime}\subset D$ we have $v\leq u$ in $D^{\prime}$.

Theorem 3.3**.**

Assume $\bf F1^{\circ}-F2^{\circ}$ hold. Let $D$ be a bounded $C^{2,\alpha}$ domain and $g\in C^{2,\alpha}(\overline{D})$. There exists a viscosity solution $u$ to

[TABLE]

such that $u\in W^{2,p}(D)$ for every $p\geq 1$.

Proof.

We use a standard penalization argument [Friedman]. Let $\beta_{\varepsilon}(t),t\in\mathbb{R}$ be a family of $C^{\infty}$ functions such that

[TABLE]

Given $\varepsilon>0$, there is a solution $v$ of

[TABLE]

Observe that Perron’s method implies that for every $\varepsilon>0$ the maximal solution $u_{\varepsilon}$ exists. Furthermore, since $\beta_{\varepsilon}$ are uniformly bounded then $\|v\|_{W^{2,p}(D)}\leq C$ with some $C$ independent of $\varepsilon$, see Theorem 7.1 [CC-Fully] and Remark 2.2 above.

If $v$ is a subsolution, i.e. $F(D^{2}v)\geq-\chi_{\{v>0\}}$ then by (3.2) we also have $F(D^{2}v)\geq-\beta_{\varepsilon}(v)$. Thus for $\varepsilon>\varepsilon^{\prime}$ (using (3.2)) we get

[TABLE]

This shows that $u_{\varepsilon^{\prime}}$ is a subsolution to (3.3). Since $u_{\varepsilon}$ is the maximal solution then we have

[TABLE]

Thus $u(x)=\lim_{\varepsilon\to 0}u_{\varepsilon}$ in $W^{2,p}$. From the uniform convergence it follows that $u(z)>0$ implying that $u_{\varepsilon}>0$ in some neighborhood of $z$. Thus $F(D^{2}u)=-1$ near $z$. Since $D^{2}u=0$ a.e. on $\{u=0\}$ it follows that $F(D^{2}u)=-\chi_{\{u>0\}}$. ∎

3.2. Non-degeneracy

Theorem 3.4**.**

Assume $\bf F1^{\circ}-F2^{\circ}$ hold. Let $u$ be the maximal solution, then there is a universal constant $c_{n,\gamma}$, depending only on dimension $n$ and $\gamma=\frac{\Lambda(n-1)}{\lambda}-1$, such that

[TABLE]

implies that $u(x_{0})>0$.

Proof.

Let us consider

[TABLE]

where

[TABLE]

and the constant $C$ is chosen so that $b(x)$ is $C^{1}$ regular. It is straightforward to compute $D^{2}b$ and thus

[TABLE]

From the ellipticity (2.1) we get that

[TABLE]

Hence

[TABLE]

Consequently, we see that $\hat{b}(x)=\frac{b(x)}{2CF(\delta_{ij})}$ is a subsolution.

Given $r$, choose $\rho$ so that $\frac{2}{\rho}=r$. Then for $|x|>\frac{1}{\rho}$ we have

[TABLE]

and consequently

[TABLE]

Thus $u(0)\geq\hat{b}(0)>0$.

∎

4. Dichotomy

In order to formulate the main result of this section we first introduce the notion of rank-2 flatness. Let $P_{2}$ be the set of all homogeneous normalized polynomials of degree two, i.e.

[TABLE]

where $a_{ij}$ is a symmetric $n\times n$ matrix. For given $p\in P_{2}$ and $x_{0}\in\mathbb{R}^{n}$, we set $p_{x_{0}}(x):=p(x-x_{0})$ and consider the zero level set of translated polynomial $p$

[TABLE]

By definition $S(p,x_{0})$ is a cone with vertex at $x_{0}$.

Definition 4.1**.**

Let $\delta>0$, $R>0$ and $x_{0}\in\partial\{u>0\}$. We say that $\partial\{u>0\}$ is $(\delta,R)$-rank-2 flat at $x_{0}$ if, for every $r\in(0,R]$, there exists $p\in P_{2}$ such that

[TABLE]

Here ${\rm{HD}}$ denotes the Hausdorff distance defined as follows

[TABLE]

Given $r>0$, $x_{0}\in\partial\{u>0\}$ and $p\in P_{2}$, we let

[TABLE]

Then, we define the rank-2 flatness at level $r>0$ of $\partial\{u>0\}$ at $x_{0}$ as follows.

Definition 4.2**.**

Let $\delta>0$, $r>0$ and $x_{0}\in\partial\{u>0\}$. We say that $\partial\{u>0\}$ is $\delta$-rank-2 flat at level $r$ at $x_{0}$ if $h(r,x_{0})<\delta r$, where

[TABLE]

In view of Definitions 4.1 and 4.2, we can say that $\partial\{u>0\}$ is $(\delta,R)$-rank-2 flat at $x_{0}\in\partial\{u>0\}$ if and only if, for every $r\in(0,R]$, it is $\delta$-rank-2 flat at level $r$ at $x_{0}$.

Theorem 4.3**.**

Let $n\geq 2$ and $u$ be a viscosity solution of  (1.1). Let $D\subset\Omega$, $\delta>0$ and let $x_{0}\in\partial\{u>0\}\cap D$ such that $|\nabla u(x_{0})|=0$ and $\partial\{u>0\}$ is not $\delta$-rank-2 flat at $x_{0}$ at any level $r>0$. Then, $u$ has at most quadratic growth at $x_{0}$, bounded from above in dependence on $\delta$.

Theorem 4.3 will follow from Proposition 4.4 below in standard way, see [DK]. Let us define $r_{k}=2^{-k}$ and $M\left(r_{k},x_{0}\right)=\sup_{B_{r_{k}}\left(x_{0}\right)}\left|u\right|,$ where $x_{0}\in\partial\left\{u>0\right\}\cap\left\{\left|\nabla u\right|=0\right\}$.

Proposition 4.4**.**

Let $u$ be as in Theorem 4.3 and $\sup|u|\leq 1$. If

[TABLE]

for some $\delta>0$, then there exists $C=C(\delta,n,\lambda,\Lambda)$ such that

[TABLE]

Proof.

If (4.6) fails then there are solutions $\{u_{j}\}$ of (1.1) with $\sup|u_{j}|\leq 1$, sequences $\{k_{j}\}$ of integers, and free boundary points $\{x_{j}\},x_{j}\in B_{1}$ such that

[TABLE]

where with some abuse of notation we set $M\left(r_{k_{j}},x_{j}\right)=\sup_{B_{r_{k_{j}}}(x_{j})}|u_{j}|$. Since $M(r_{k_{j}},x_{j})\leq\sup_{B_{1}}|u_{j}|<\infty$ it follows that $k_{j}\to\infty$. Define the scaled functions

[TABLE]

By construction we have

[TABLE]

where the last inequality follows from (4.7) after rescaling the inequality $\frac{M(r_{k_{j}-m},x_{j})}{M(r_{k_{j}+1},x_{j})}<2^{2(m-1)}$. Utilizing the homogeneity of operator $F$ and noting that

[TABLE]

it follows that

[TABLE]

where $\sigma_{j}=\frac{r_{k_{j}}^{2}}{M(r_{k_{j}+1},x_{j})}$. Observe that $\sigma_{j}<\frac{1}{j}$ in view of (4.7). Since under hypotheses $\bf F1^{\circ}-F2^{\circ}$ we have local $W^{2,p}$ bounds for all $p\geq 1$ (see Theorem 7.1 [CC-Fully]) it follows that we can employ a customary compactness argument for the viscosity solutions to show that there is a function $v_{0}\in W^{2,p}_{loc}(\mathbb{R}^{n})$ such that

[TABLE]

From Liouville’s theorem it follows that $v_{0}$ is homogeneous quadratic polynomial of degree two. This is in contradiction with (4.9) and the proof is complete. ∎

Remark 4.5**.**

In [Lee-Sh] the authors proved some partial results for the problem

[TABLE]

For $F=\Delta$ this problem arises in the linear potential theory related to harmonic continuation of the Newtonian potential of $B_{1}\cap\mathcal{D}$.

Analysis similar to that in the proof of Proposition 4.4 shows that the result is also valid for the solutions of (4.10).

Corollary 4.6**.**

Let $u$ be a viscosity solution to (4.10), then the statement of Theorem 4.3 holds for $u$ too.

5. Quadruple junctions

Throughout this section we assume that $F$ is convex, satisfies $\bf F1^{\circ}-F2^{\circ}$ and $u$ is a viscosity solution, see Section 3.

Lemma 5.1**.**

Assume $\bf F1^{\circ}-F2^{\circ}$ hold and $F$ is convex. Let $n=2$ and $0\in\partial\{u>0\},|\nabla u(0)|=0$ be a rank-2 flat point such that the zero set of the polynomial $p(x)=M^{2}x_{1}^{2}-x_{2}^{2},M>0$ approximates $\partial\{u>0\}$ near [math]. Assume further that $u$ is non-degenerate at [math]. Then for every $\delta_{0}>0$ there is $r_{0}=2^{-k_{0}}$ (for some $k_{0}\in\mathbb{N}$) such that $\partial_{2}u(x+te)\geq 0$ whenever $x\in(B_{r_{0}}\setminus B_{\delta_{0}r_{0}})\cap K^{-}$ and $\delta_{0}\leq t\leq 2$.

Proof.

Let $\theta_{0}=\arctan M$ and denote $K^{-}=\{x_{2}\geq M|x_{1}|\}$. After rotation of coordinate system we can assume that $K^{-}$ contains $u<0$ away from some small neighborhood of $x_{2}=M|x_{1}|$ (the green cones in Figure 1 represent that neighborhood).

Suppose the claim fails, then there is $\delta_{0}>0$ so that for every $r_{k}=2^{-k}\to 0$ and some points $x_{k}\in B_{r_{k}}\setminus B_{\delta_{0}r_{k}}\cap\Omega^{-}(u)$ we have

[TABLE]

We can choose $\delta_{0}$ so that for large $k$ there holds $\delta_{0}>\frac{h_{k}}{\cos\theta_{0}}\to 0,$ where $h_{k}=h(2^{-k},0)$.

Introduce the scaled functions

[TABLE]

Here we set $M(r_{k})=M(r_{k},0)$. For both scalings we have that $v_{k}$’s are non-degenerate: for the first scaling it follows from Theorem 3.4 (our assuption on non-degeneracy), for the second one it follows that $\sup_{B_{1/2}}|v_{k}|=1$.

Moreover, by (5.1) there is $y_{k}\in(B_{1}\setminus B_{\delta_{0}})\cap\{v_{k}<0\}$ such that

[TABLE]

There is a subsequence $y_{k_{j}}+t_{k_{j}}e_{2}\to y_{0}+t_{0}e_{2}\in K^{-}\cap B_{2},$ there is a Harnack chain $B^{1},\dots,B^{N}$ where $B^{1}=B_{\cos\theta_{0}/2}(e_{2})$ and $B^{N}=B_{\delta_{0}/2}(y_{0})$, $N$ is independent of $k_{j}$. Let $\widetilde{K}=B_{1}\cup\bigcup_{i=1}^{N}B^{i}$. Since under hypotheses $\bf F1^{\circ}-F2^{\circ}$ we have local $W^{2,p}$ bounds for all $p\geq 1$ (see Theorem 7.1 [CC-Fully]) it follows that we can employ a customary compactness argument for viscosity solutions to infer that there is a function $v_{0}\in W^{2,p}_{loc}(\mathbb{R}^{n})$ such that we have

[TABLE]

From Theorem 2.6 it follows that $\partial_{2}v_{0}(y_{0}+t_{0}e_{2})=0$. Moreover, $w=\partial_{2}v_{0}$ satisfies the equation $F_{ij}D_{ij}w=0$ in $\widetilde{K}$, hence from the strong maximum principle it follows that $w=0$ in $K^{-}$. Consequently, $v_{0}$ depends only on $x_{1}$ implying that $\theta_{0}=0$ or $\theta_{0}=\pi/2$ which is a contradiction.

∎

Theorem 5.2**.**

Let $u$ be as in Lemma 5.1. Then in some neighbourhood of [math] the free boundary consists of four $C^{1}$ curves tangential to the zero set of the polynomial $M^{2}x_{1}^{2}-x_{2}^{2}$.

Proof.

Let $\partial_{\rm{sing}}\{u>0\}=\partial\{u>0\}\cap\{|\nabla u|=0\}$. Clearly, it is enough to prove that there is $r$ such that $\partial_{\rm{sing}}\{u>0\}\cap B_{r}=\{0\}$. Suppose the claim fails. Then there is a sequence $x_{k}\in\partial_{\rm{sing}}\{u>0\}$, $x_{k}\to 0$. Let $M_{k}^{-}:=M^{-}(2r_{k}\ell_{0})=\sup_{B_{2r_{k}\ell_{0}}}u^{-},r_{k}=|x_{k}|$ and consider

[TABLE]

Note that $F(D^{2}v_{k})=-\chi_{\{v_{k}>0\}}\frac{r_{k}^{2}}{M^{-}(2r_{k}\ell_{0})}$, and therefore by dichotomy (see Section 4) and non-degeneracy $|F(D^{2}v_{k})|\leq C$ for some $C>0$ independent of $k$.

By construction $\sup_{B_{2\ell_{0}}}|v_{k}^{-}|=1$ and since $F(D^{2}v_{k})=0$ in $\Omega^{-}(v_{k}):=\{v_{k}<0\}$ it follows that there is $z_{k}\in\partial B_{2\ell_{0}}\cap\Omega^{-}(v_{k})$ such that $v_{k}^{-}(z_{k})=1$. Consequently, $\operatorname{dist}(z_{k}+\delta_{0}e_{2},\{p=0\})\geq\delta_{0}/2$ and by Lemma 5.1

[TABLE]

Claim 5.3**.**

With the notation above we have

[TABLE]

To check this we first observe that $\text{trace}(A_{t}D^{2}v_{k}(x))\leq F(D^{2}v_{k}x)$ thanks to the convexity and $A_{t}\in\mathcal{S}_{\lambda,\Lambda}$. Now consider $w_{k,t}(x)=v_{k}(A^{\frac{1}{2}}_{t}x)$ then $\Delta w_{k,t}(x)=\text{trace}(A_{t}D^{2}v_{k}(x))\leq F(D^{2}v_{k}(x))$. Since $w_{k,t}$ is continuous and $w_{k,t}(0)=0$ then one can easily check that

[TABLE]

because of convexity of $F$ and the estimate $F(D^{2}v_{k})\leq 0$.

Note that

[TABLE]

Thus from (5.5) it follows that

[TABLE]

Consequently, we get

[TABLE]

Let $\widehat{v}_{k}=v_{k}+\widehat{C}|x|^{2}$ then

[TABLE]

provided that $\widehat{C}$ is sufficiently large.

We see that $\widehat{v}_{k}$ is a subsolution, and hence so is $\widehat{v}_{k}^{+}$. Consequently, applying the weak Harnack inequality [CC-Fully] we get

[TABLE]

This completes the proof of the claim.

Thus, as in the proof of Lemma 5.1, we can employ a customary compactness argument in $W^{2,p}$ so that $y_{k}=x_{k}/r_{k}\to y_{0}\in\{x_{2}=M|x_{1}|\}\cap\partial B_{1}$ and

[TABLE]

by Harnack chain and $C^{1,\alpha}$ estimates in the Harnack chain domain (which joins $2\ell_{0}e_{2}$ with $z_{0}+\delta_{0}e_{2}$). Since at $y_{0}$ free boundary is a line we can apply Hopf’s lemma to conclude that $v_{0}^{-}\equiv 0$ which is a contradiction. ∎

6. Existence of homogenous blow-ups

In this section we show that if the free boundary is a cone then one can blow-up the solution at the vertex so that the limit is a homogeneous function. We start with a doubling inequality which provides a bound for the rate of the scaling at the vertex.

Lemma 6.1**.**

Assume $\bf F1^{\circ}-F2^{\circ}$ hold and $F$ is concave. Let $\mathcal{K}_{t}=\{x=r\sigma,\sigma\in S,0<r<t\}$, $S\subset\mathbb{S}^{n-1}$ such that $\partial S$ is smooth. Let $v$ be the solution to the Dirichlet problem $F(D^{2}v)=0$ in $\mathcal{K}_{1}$ and $v=v_{0}$ on $\partial\mathcal{K}_{1}$ where

[TABLE]

Then there is a constant $\varepsilon>0$ such that

[TABLE]

Proof.

Existence of $v$ follows from the Perron’s method [CC-Fully].

Consider the barrier $b\left(x\right)=1+\dfrac{1}{\alpha^{2}}\left(e^{-\alpha}-e^{-\alpha\left|x\right|^{2}}\right)$ for some $\alpha>0$ to be fixed below. We have

[TABLE]

Consequently, in the ring $\frac{1}{2}\leq|x|\leq 1$ we have from $\bf F1^{\circ}$ and $\bf F2^{\circ}$ that

[TABLE]

provided that $\alpha\geq 2\frac{\Lambda}{\lambda}$.

Since $b$ is concave function of $r=|x|$ and $v_{0}$ is convex in $r$ such that $b(0)\geq v_{0}(0)$ and $b(x)\geq v(x),|x|=1$, then it follows that $b(x)\geq v_{0}(x)$ on $\partial(\mathcal{K}_{1}\setminus K_{\frac{1}{2}})$. Moreover, by the maximum principle we have $v(x)\leq 1$ on $\partial K_{\frac{1}{2}}\cap\partial B_{\frac{1}{2}}$. Thus $v(x)\leq b(x)$ on $\partial(\mathcal{K}_{1}\setminus K_{\frac{1}{2}})$, which in conjunction with $F(D^{2}b(x))\leq 0,\frac{1}{2}\leq|x|\leq 1$ and the comparison principle implies that $v(x)\leq b(x)$ in $(\mathcal{K}_{1}\setminus K_{\frac{1}{2}})$.

Furthermore, for $x\in\partial\mathcal{K}_{1}\setminus S$ we have $v(Rx)=0$, hence (6.1) is true for every $x\in\partial\mathcal{K}_{1}\setminus S$. On the other hand on $S$ (where $|x|=1$) we have

[TABLE]

where the last line follows from the mean value theorem. Thus (6.1) holds on $\partial\mathcal{K}_{1}$ with $\varepsilon=\frac{e^{-\alpha}}{\alpha}$. Applying the comparison principle to $v(Rx)$ and $(1-\varepsilon(1-R))v(x)$ the result follows. ∎

With the help of Lemma 6.1 we can prove the existence of a homogeneous solution of the form $r^{\kappa}\phi(\sigma)$ vanishing on the boundary of the cone $\mathcal{K}_{1}$.

Theorem 6.2**.**

There is $\kappa>0$ and $\phi\in C^{\infty}(\overline{S})$ satisfying $F(D^{2}(r^{\kappa}\phi(\sigma)))=0,\phi|_{\partial S}=0,\phi|_{K_{1}}>0$.

Proof.

First we want to compare $v$ with its scalings in order to obtain two sided bounds. From (6.1), the strong maximum principle and Hopf’s lemma (see [Giga]), we derive that there is a $k>0$ such that

[TABLE]

for all $x\in\partial\mathcal{K}_{\frac{1}{4}}$. This and the weak comparison principle imply that (6.3) holds for all $x\in\mathcal{K}_{\frac{1}{4}}$. Consequently, we can use the $C^{0}$-estimate of [CC-Fully] and (6.1) (combined with the barrier argument in [Wang]) in order to obtain a $v^{*}\in C^{0}(\overline{\Omega})\cap C^{1}(\Omega\setminus\{0\})$ and a sequence of $R_{k}>0$ tending to zero such that

[TABLE]

in the sense $C^{0}(\overline{\Omega})\cap C^{1}(\Omega\setminus\{0\})$. Moreover, $v^{*}\geq 0$ in $\mathcal{K}_{1}$, $v^{*}=0$ on $\partial\mathcal{K}_{1}\setminus S$, $\|v^{*}\|_{L^{\infty}(\mathcal{K}_{1})}=1$ and $v^{*}$ is a viscosity solution of $F(D^{2}v^{\ast})=0$ in $\mathcal{K}_{1}$. By the strong maximum principle (see Lemma 2.3) and Hopf’s lemma [Giga], $v^{*}>0$, in $\mathcal{K}_{1}$, and $\nabla v^{*}\not=0$ on $\partial\mathcal{K}_{1}\setminus\{\{0\}\cup\overline{S}\}$.

Let $R\in(0,1)$. By Lemma 2.7 it is enough to prove the existence of a $C_{R}>0$ satisfying

[TABLE]

In order to prove (6.5) let us define $E_{r,R}=\{c:cu(x)\leq u(Rx),\forall x\in\mathcal{K}_{r}\}$. By (6.3) $E_{r,R}\not=\emptyset$. Note that $E_{r_{2},R}\subset E_{r_{1},R}$, if $r_{1}<r_{2}$ and thus

[TABLE]

In order to show (6.5), we set

[TABLE]

By the weak comparison principle, $c_{r,R}$ is decreasing with respect to $r\in(0,1]$. This and (6.4) imply that

[TABLE]

Let us suppose that (6.5) is not true. Then, we can use (6.6), the strong comparison principle and Hopf’s comparison principle (see Section 2, Lemmata 2.3 and 2.4) to obtain a $\delta>0$ such that

[TABLE]

This and (6.4) show that

[TABLE]

for some $r>0$. By the weak comparison principle, (6.7) holds for all $x\in\mathcal{K}_{r}$. This, however, is a contradiction to the definition of $C_{R}$. Hence, (6.5) must be true. ∎

The homogeneous solutions $r^{\kappa}\phi(\sigma)$, constructed in Theorem 6.2, provide two-sided control for the scalings of the solutions of $F(D^{2}u)=0$ in the cone $\mathcal{K}_{1}$.

Theorem 6.3**.**

Let $u\geq 0$ be a viscosity solution of $F(D^{2}u)=0$ in the cone $\mathcal{K}_{1}$. Then then for every sequence $\{R_{k}\}_{k=1}^{\infty}$, $R_{k}\downarrow 0$ there exists a subsequence $R_{k_{j}}$ such that the functions

[TABLE]

converges locally uniformly to $r^{\kappa}\phi(\sigma)$.

Proof.

Let $b=r^{\kappa}\phi(\sigma)$ then by the boundary Harnack principle (2.4) there is a constant $C>0$ such that

[TABLE]

We want to prove that there is a subsequence of $\{R_{k}\}$ such that

[TABLE]

in $C^{0}(\overline{\mathcal{K}_{1}})\cap C^{1}(\overline{\mathcal{K}_{1}}\setminus\{0\})$.

Define

[TABLE]

By (6.8) $E_{R}\not=\emptyset$. From the weak comparison principle (as in the proof of Theorem 6.2), one derives that $c_{R}$ is decreasing with respect to $R$. Consequently, the limit

[TABLE]

exists and it is positive. The $C^{\alpha}$-estimates of [CC-Fully], $C^{1}(\overline{\mathcal{K}_{1}}\setminus\{0\})$ regularity result and (6.8) imply that there is a subsequence of $\{R_{k}\}$ and a $u^{*}\in C^{0}(\overline{\mathcal{K}_{1}})\cap C^{1}(\overline{\mathcal{K}_{1}}\setminus\{0\})$ such that

[TABLE]

in the sense of $C^{0}(\overline{\mathcal{K}_{1}})\cap C^{1}(\overline{\mathcal{K}_{1}}\setminus\{0\})$. Moreover, $u^{*}=0$ on $\mathcal{K}_{1}\setminus S$, $u^{*}$ solves $F(D^{2}u^{*})=0$ in $\mathcal{K}_{1}$ and

[TABLE]

Now, suppose that $u^{*}$ is not identical to $c^{*}b$ , in $\mathcal{K}_{1}$. Then, (6.11), the strong comparison principle and Hopf’s comparison principle (see Section 2, Lemmata 2.3 and 2.4) imply that there is a $\delta>0$ such that

[TABLE]

This and (6.10) show that

[TABLE]

for some $r>0$. The weak comparison principle shows that this is true, also in $\mathcal{K}_{r}$. This, however, is a contradiction to the definition of $c^{*}$. Hence, $c^{*}b=u^{*}$, in $\mathcal{K}_{1}$, and (6.9) is true. ∎