Existence of viscosity multi-valued solutions with asymptotic behavior for Hessian equations

Abstract

The Perron method is used to establish the existence of viscosity multi-valued solutions for a class of Hessian-type equations with prescribed behavior at infinity.

1 Introduction

In [1], [2], the multi-valued solutions of the eikonal equation were studied. Later, in [3], [4] Jin et al. provided a level set method for the computation of multi-valued geometric solutions to general quasilinear partial differential equations and multi-valued physical observables to the semiclassical limit of the Schrödinger equations. In [5], Caffarelli and Li investigated the multi-valued solutions of the Monge-Ampère equation where they first introduced the geometric situation of the multi-valued solutions and obtained the existence, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. In [6] Ferrer et al. used complex variable methods to study the multi-valued solutions for the Dirichlet problems of Monge-Ampère equations on exterior planar domains. Recently, Bao and Dai discussed the multi-valued solutions of Hessian equations, see [7], [8]. Motivated by the above works, in this paper we study the viscosity multi-valued solutions of the Hessian equation

$$F\left(\lambda \left(D^{2}u\right)\right)=\sigma >0,$$

(1.1)

where $\sigma$ is a constant and $\lambda (D^{2}u)=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$ are eigenvalues of the Hessian matrix $D^{2}u$. $F$ is assumed to be defined in the symmetric open convex cone $\mathrm{\Gamma }$, with vertex at the origin, containing

$${\mathrm{\Gamma }}^{+}=\{\lambda \in {\mathbf{R}}^n:\text{each component of }\lambda ,{\lambda }_i>0,i=1,2,\dots ,n\},$$

satisfies the fundamental structure conditions

$$F_i(\lambda )=\frac{\partial F}{\partial {\lambda }_i}>0\text{in }\mathrm{\Gamma },1\le i\le n,$$

(1.2)

and $F$ is a continuous concave function. In addition, $F$ will be assumed to satisfy some more technical assumptions such as

$$F>0\text{in }\mathrm{\Gamma },F=0\text{on }\partial \mathrm{\Gamma },$$

(1.3)

and for any $r\ge 1$, $R>0$,

$$F\left(R(\frac{1}{r^{n-1}},r,\dots ,r)\right)\ge F(R(1,1,\dots ,1)).$$

(1.4)

For every $C>0$ and every compact set $K$ in $\mathrm{\Gamma }$, there is $\mathrm{\Lambda }=\mathrm{\Lambda }(C,K)$ such that

$$F(\mathrm{\Lambda }\lambda )\ge C\text{for all }\lambda \in K.$$

(1.5)

There exists a number $\mathrm{\Lambda }$ sufficiently large such that at every point $x\in \partial \mathrm{\Omega }$, if $x_1,\dots ,x_{n-1}$ represent the principal curvatures of $\partial \mathrm{\Omega }$, then

$$(x_1,\dots ,x_{n-1},\mathrm{\Lambda })\in \mathrm{\Gamma }.$$

(1.6)

Inequality (1.4) is satisfied by each $k$th root of an elementary symmetric function ($1\le k\le n$) and the $(k-l)$th root of each quotient of the $k$th elementary symmetric function and the $l$th elementary symmetric function ($1\le l<k\le n$).

2 Preliminaries

The geometric situation of the multi-valued function is given in [5]. Let $n\ge 2$, $D\subset {\mathbf{R}}^n$ be a bounded domain with smooth boundary $\partial D$, and let $\mathrm{\Sigma }\subset D$ be homeomorphic in ${\mathbf{R}}^n$ to an $n-1$ dimensional closed disc. $\partial \mathrm{\Sigma }$ is homeomorphic to an $n-2$ dimensional sphere for $n\ge 3$.

Let $\mathbf{Z}$ be the set of integers and $M=(D\setminus \partial \mathrm{\Sigma })\times \mathbf{Z}$ denote a covering of $D\setminus \partial \mathrm{\Sigma }$ with the following standard parametrization: fixing $x^{\ast }\in D\setminus \partial \mathrm{\Sigma }$ and connecting $x^{\ast }$ by a smooth curve in $D\setminus \partial \mathrm{\Sigma }$ to a point $x$ in $D\setminus \partial \mathrm{\Sigma }$. If the curve goes through $\mathrm{\Sigma }$$m\ge 0$ times in the positive direction (fixing such a direction), then we arrive at $(x,m)$ in $M$. If the curve goes through $\mathrm{\Sigma }$$m\ge 0$ times in the negative direction, then we arrive at $(x,-m)$ in $M$.

For $k=2,3,\dots$ , we introduce an equivalence relation ‘$\sim k$’ on $M$ as follows: $(x,m)$ and $(y,j)$ in $M$ are ‘$\sim k$’ equivalent if $x=y$ and $m-j$ is an integer multiple of $k$. We let $M_k=M/\sim k$ denote the $k$-sheet cover of $D\setminus \partial \mathrm{\Sigma }$, and let ${\partial }^{\prime }{M}_k={\bigcup }_{m=1}^k(\partial D\times \{m\})$.

We define a distance in $M_k$ as follows: for any $(x,m),(y,j)\in M_k$, let $l((x,m),(y,j))$ denote a smooth curve in $M_k$ which connects $(x,m)$ and $(y,j)$, and let $|l((x,m),(y,j))|$ denote its length. Define

$$d((x,m),(y,j))=\underset{l}{inf}\left|l((x,m),(y,j))\right|,$$

where the infimum is taken over all smooth curves connecting $(x,m)$ and $(y,j)$. Then $d((x,m),(y,j))$ is a distance.

Definition 2.1

We say that a function $u$ is continuous at $(x,m)$ in $M_k$ if

$$\underset{d((x,m),(y,j))\to 0}{lim}u(y,j)=u(x,m),$$

and $u\in C^0(M_k)$ if for any $(x,m)$, $u$ is continuous at $(x,m)$.

Similarly, we can define $u\in C^{\alpha }(M_k)$, $C^{0,1}(M_k)$ and $C^2(M_k)$.

Definition 2.2

A function $u\in C^2(M_k)$ is called admissible if $\lambda \in \overline{\mathrm{\Gamma }}$, where $\lambda =\lambda (D^{2}u(x,m))=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$ are the eigenvalues of the Hessian matrix $D^{2}u(x,m)$.

Definition 2.3

A function $u\in C^0(M_k)$ is called a viscosity subsolution (resp. supersolution) to (1.1) if for any $(y,m)\in M_k$ and $\xi \in C^2(M_k)$ satisfying

$$u(x,m)\le (\text{resp.}\ge )\xi (x,m),(x,m)\in M_k\text{and}u(y,m)=\xi (y,m),$$

we have

$$F\left(\lambda (D^2\xi (y,m))\right)\ge (\text{resp.}\le )\sigma .$$

Definition 2.4

A function $u\in C^0(M_k)$ is called a viscosity solution to (1.1) if it is both a viscosity subsolution and a viscosity supersolution to (1.1).

Definition 2.5

A function $u\in C^0(M_k)$ is called admissible if for any $(y,m)\in M_k$ and any function $\xi \in C^2(M_k)$ satisfying $u(x,m)\le (\ge )\xi (x,m)$, $x\in M_k$, $u(y,m)=\xi (y,m)$, we have $\lambda (D^2\xi (y,m))\in F$.

Remark

It is obvious that if $u$ is a viscosity subsolution, then $u$ is admissible.

Lemma 2.1

Let$\mathrm{\Omega }$be a bounded strictly convex domain in${\mathbf{R}}^n$, $\partial \mathrm{\Omega }\in C^2$, $\phi \in C^2(\overline{\mathrm{\Omega }})$. Then there exists a constant$C$only dependent on$n$, $\phi$and$\mathrm{\Omega }$such that for any$\xi \in \partial \mathrm{\Omega }$, there exists$\overline{x}(\xi )\in {\mathbf{R}}^n$such that

$$|\overline{x}(\xi )|\le C,w_{\xi }(x)<\phi (x)\mathit{\text{for }}x\in \overline{\mathrm{\Omega }}\setminus \{\xi \},$$

where$w_{\xi }(x)=\phi (\xi )+\frac{\overline{R}}{2}({|x-\overline{x}(\xi )|}^2-{|\xi -\overline{x}(\xi )|}^2)$for$x\in {\mathbf{R}}^n$and$\overline{R}$is a constant satisfying$F(\overline{R},\overline{R},\dots ,\overline{R})=\sigma$.

This is a modification of Lemma 5.1 in [5].

Lemma 2.2

Let$\mathrm{\Omega }$be a domain in${\mathbf{R}}^n$and$f\in C^0({\mathbf{R}}^n)$be nonnegative. Assume that the admissible functions$v\in C^0(\overline{\mathrm{\Omega }})$, $u\in C^0({\mathbf{R}}^n)$satisfy, respectively,

$$\begin{array}{c}F\left(\lambda \left(D^{2}v\right)\right)\ge f(x),x\in \mathrm{\Omega },\hfill \\ F\left(\lambda \left(D^{2}u\right)\right)\ge f(x),x\in {\mathbf{R}}^n.\hfill \end{array}$$

Moreover,

$$\begin{array}{c}u\le v,x\in \overline{\mathrm{\Omega }},\hfill \\ u=v,x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

Set

$$w(x)=\{\begin{array}{ll}v(x),& x\in \mathrm{\Omega },\\ u(x),& x\in {\mathbf{R}}^n\setminus \mathrm{\Omega }.\end{array}$$

Then $w\in C^0({\mathbf{R}}^n)$ is an admissible function and satisfies in the viscosity sense

$$F\left(\lambda (D^{2}w(x))\right)\ge f(x),x\in {\mathbf{R}}^n.$$

Lemma 2.3

Let$B$be a ball in${\mathbf{R}}^n$and let$f\in C^{0,\alpha }(\overline{B})$be positive. Suppose that$\underline{u}\in C^0(\overline{B})$satisfies in the viscosity sense

$$F\left(\lambda \left(D^{2}u\right)\right)\ge f(x),x\in B.$$

Then the Dirichlet problem

$$\begin{array}{c}F\left(\lambda \left(D^{2}u\right)\right)=f(x),x\in B,\hfill \\ u=\underline{u}(x),x\in \partial B\hfill \end{array}$$

admits a unique admissible viscosity solution$u\in C^0(\overline{B})$.

We refer to [9] for the proof of Lemmas 2.2 and 2.3.

3 Existence of viscosity multi-valued solutions with asymptotic behavior

In this section, we establish the existence of viscosity multi-valued solutions with prescribed asymptotic behavior at infinity of (1.1). Let $\mathrm{\Omega }$ be a bounded strictly convex domain with smooth boundary $\partial \mathrm{\Omega }$. Let $\mathrm{\Sigma }$, diffeomorphic to an $(n-1)$-disc, be the intersection of $\mathrm{\Omega }$ any hyperplane in ${\mathbf{R}}^n$. Let $M=({\mathbf{R}}^n\setminus \partial \mathrm{\Sigma })\times \mathbf{Z}$, $M_k=M/\sim k$ be covering spaces of ${\mathbf{R}}^n\setminus \partial \mathrm{\Sigma }$ as in Section 2. $\mathrm{\Sigma }$ divides $\mathrm{\Omega }$ into two open parts, denoted as ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$. Fixing $x^{\ast }\in {\mathrm{\Omega }}^{-}$, we use the convention that going through $\mathrm{\Sigma }$ from ${\mathrm{\Omega }}^{-}$ to ${\mathrm{\Omega }}^{+}$ denotes the positive direction through $\mathrm{\Sigma }$. Our main result is the following theorem.

Theorem 3.1

Let$k\ge 3$. Then, for any$C_m\in \mathbf{R}$, there exists an admissible viscosity solution$u\in C^0(M_k)$of

$$F\left(\lambda \left(D^{2}u\right)\right)=\sigma ,(x,m)\in M_k$$

(3.1)

satisfying

$$\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{|x|}^{n-2}|u(x,m)-(\frac{\overline{R}}{2}{|x|}^2+C_m)|<+\mathrm{\infty },$$

(3.2)

where$\overline{R}$is a constant satisfying$F(\overline{R},\overline{R},\dots ,\overline{R})=\sigma$.

When

$$F\left(\lambda \left(D^{2}u\right)\right)={\sigma }_k\left(\lambda \left(D^{2}u\right)\right),\mathrm{\Gamma }={\mathrm{\Gamma }}_k=\{\lambda \in {\mathbf{R}}^n:{\sigma }_j>0,j=1,2,\dots ,k\},$$

where the $k$th elementary symmetric function

$${\sigma }_k(\lambda )=\sum _{i_1<\cdots <i_k}{\lambda }_{i_1}\cdots {\lambda }_{i_k}$$

for $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$, in [8] Dai obtained the following result.

Theorem 3.2

Let$k\ge 3$. Then, for any$C_m\in \mathbf{R}$, there exists a$k$-convex viscosity solution$u\in C^0(M_k)$of

$${\sigma }_k\left(\lambda \left(D^{2}u\right)\right)=1,(x,m)\in M_k$$

satisfying

$$\underset{|x|\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\left({|x|}^{k-2}\right|u(x,m)-(\frac{C_{\ast }}{2}{|x|}^2+C_m)\left|\right)<\mathrm{\infty },$$

where$C_{\ast }={(\frac{1}{C_n^k})}^{\frac{1}{k}}$.

Proof of Theorem 3.1

We divide the proof of Theorem 3.1 into two steps.

Step 1. By [10], there is an admissible solution $\mathrm{\Phi }\in C^{\mathrm{\infty }}(\overline{\mathrm{\Omega }})$ of the Dirichlet problem:

$$\begin{array}{c}F\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)=C_0>\sigma ,x\in \mathrm{\Omega },\hfill \\ \mathrm{\Phi }=0,x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

By the comparison principles in [11], $\mathrm{\Phi }\le 0$ in $\mathrm{\Omega }$. Further, by Lemma 2.1, for each $\xi \in \partial \mathrm{\Omega }$, there exists $\overline{x}(\xi )\in {\mathbf{R}}^n$ such that

$$W_{\xi }(x)<\mathrm{\Phi }(x),x\in \overline{\mathrm{\Omega }}\setminus \{\xi \},$$

where

$$W_{\xi }(x)=\frac{\overline{R}}{2}({|x-\overline{x}(\xi )|}^2-{|\xi -\overline{x}(\xi )|}^2),\xi \in {\mathbf{R}}^n,$$

and ${sup}_{\xi \in \partial \mathrm{\Omega }}|\overline{x}(\xi )|<\mathrm{\infty }$. Therefore

$$\begin{array}{c}{W}_{\xi }(\xi )=0,W_{\xi }(x)\le \mathrm{\Phi }(x)\le 0,x\in \overline{\mathrm{\Omega }},\hfill \\ F\left(\lambda (D^2{W}_{\xi }(x))\right)=F(\overline{R},\overline{R},\dots ,\overline{R})=\sigma ,\xi \in {\mathbf{R}}^n.\hfill \end{array}$$

Denote

$$W(x)=\underset{\xi \in \partial \mathrm{\Omega }}{sup}{W}_{\xi }(x).$$

Then

$$W(x)\le \mathrm{\Phi }(x),x\in \mathrm{\Omega },$$

and by [12]

$$F\left(\lambda \left(D^{2}W\right)\right)\ge \sigma ,x\in {\mathbf{R}}^n.$$

Define

$$V(x)=\{\begin{array}{ll}\mathrm{\Phi }(x),& x\in \mathrm{\Omega },\\ W(x),& x\in {\mathbf{R}}^n\setminus \mathrm{\Omega }.\end{array}$$

Then $V\in C^0({\mathbf{R}}^n)$ is an admissible viscosity solution of

$$F\left(\lambda \left(D^{2}V\right)\right)\ge \sigma ,x\in {\mathbf{R}}^n.$$

Fix some $R_1>0$ such that $\overline{\mathrm{\Omega }}\subset B_{R_1}(0)$, where $B_{R_1}(0)$ is the ball centered at the origin with radius $R_1$.

Let $R_2=2R_1{\overline{R}}^{\frac{1}{2}}$. For $a>1$, defuse

$$W_a(x)=\underset{B_{R_1}}{inf}V+{\int }_{2R_2}^{|{\overline{R}}^{\frac{1}{2}}x|}{(s^n+a)}^{\frac{1}{n}}ds,x\in {\mathbf{R}}^n.$$

Then

$$D_{ij}{W}_a={({|y|}^n+a)}^{\frac{1}{n}-1}[({|y|}^{n-1}+\frac{a}{|y|})\overline{R}{\delta }_{ij}-\frac{a{\overline{R}}^2{x}_i{x}_j}{{|y|}^3}],|x|>0,$$

where $y={\overline{R}}^{\frac{1}{2}}x$. By rotating the coordinates, we may set $x=(r,0,\dots ,0)$. Therefore

$$D^2{W}_a={(R^n+a)}^{\frac{1}{n}-1}\overline{R}diag(R^{n-1},R^{n-1}+\frac{a}{R},\dots ,R^{n-1}+\frac{a}{R}),$$

where $R=|y|$. Consequently, $\lambda (D^2{W}_a)\in \mathrm{\Gamma }$ for $|x|>0$ and by (1.4)

$$F\left(\lambda \left(D^2{W}_a\right)\right)\ge F(\overline{R},\overline{R},\dots ,\overline{R})=\sigma ,|x|>0.$$

Moreover,

$$W_a(x)\le V(x),|x|\le R_1.$$

(3.3)

Fix some $R_3>3R_2$ satisfying

$$R_3{\overline{R}}^{\frac{1}{2}}>3R_2.$$

We choose $a_1>1$ such that for $a\ge a_1$,

$$W_a(x)>\underset{B_{R_1}}{inf}V+{\int }_{2R_2}^{3R_2}{(s^n+a)}^{\frac{1}{n}}ds\ge V(x),|x|=R_3.$$

Then by (3.3) $R_3\ge R_1$. According to the definition of $W_a$,

$$\begin{array}{rcl}{W}_a(x)& =& \underset{B_{R_1}}{inf}V+{\int }_{2R_2}^{|{\overline{R}}^{\frac{1}{2}}x|}s({(1+\frac{a}{s^n})}^{\frac{1}{n}}-1)ds+{\int }_{2R_2}^{|{\overline{R}}^{\frac{1}{2}}x|}sds\\ =& \frac{\overline{R}}{2}{|x|}^2+C_m+\underset{B_{R_1}}{inf}V+{\int }_{2R_2}^{+\mathrm{\infty }}s({(1+\frac{a}{s^n})}^{\frac{1}{n}}-1)ds-C_m\\ -2R_2^2-{\int }_{|{\overline{R}}^{\frac{1}{2}}x|}^{+\mathrm{\infty }}s({(1+\frac{a}{s^n})}^{\frac{1}{n}}-1)ds,x\in {\mathbf{R}}^n.\end{array}$$

Let

$$\mu (m,a)=\underset{B_{R_1}}{inf}V+{\int }_{2R_2}^{+\mathrm{\infty }}s({(1+\frac{a}{s^n})}^{\frac{1}{n}}-1)ds-C_m-2R_2^2.$$

Then $\mu (m,a)$ is continuous and monotonic increasing for $a$ and when $a\to \mathrm{\infty }$, $\mu (m,a)\to \mathrm{\infty }$, $1\le m\le k$. Moreover,

$$W_a(x)=\frac{\overline{R}}{2}{|x|}^2+C_m+\mu (m,a)-O\left({|x|}^{2-n}\right),\text{when }|x|\to \mathrm{\infty }.$$

(3.4)

Define, for $a\ge a_1$ and $1\le m\le k$,

$${\underline{u}}_{m,a}(x)=\{\begin{array}{ll}max\{V(x),W_a(x)\}-\mu (m,a),& |x|\le R_3,\\ W_a-\mu (m,a),& |x|\ge R_3.\end{array}$$

Then by (3.4), for $1\le m\le k$,

$${\underline{u}}_{m,a}(x)=\frac{\overline{R}}{2}{|x|}^2+C_m-O\left({|x|}^{2-n}\right),\text{when }|x|\to \mathrm{\infty },$$

and by the definition of $V$,

$${\underline{u}}_{m,a}(x)=-\mu (m,a),x\in \partial \mathrm{\Sigma }.$$

Choose $a_2\ge a_1$ large enough such that when $a\ge a_2$,

$$\begin{array}{rcl}V(x)-\mu (m,a)& =& V(x)-\underset{B_{R_1}}{inf}V-{\int }_{2R_2}^{+\mathrm{\infty }}s({(1+\frac{a}{s^n})}^{\frac{1}{n}}-1)ds+C_m+2R_2^2\\ \le & C_m\\ \le & \frac{\overline{R}}{2}{|x|}^2+C_m,|x|\le R_3.\end{array}$$

Therefore

$${\underline{u}}_{m,a}(x)\le \frac{\overline{R}}{2}{|x|}^2+C_m,a\ge a_2,x\in {\mathbf{R}}^n.$$

By Lemma 2.2, ${\underline{u}}_{m,a}\in C^0({\mathbf{R}}^n)$ is admissible and satisfies in the viscosity sense

$$F\left(\lambda \left(D^2{\underline{u}}_{m,a}\right)\right)\ge \sigma ,x\in {\mathbf{R}}^n.$$

It is easy to see that there exists a continuous function $a^{(m)}(a)$ such that ${lim}_{a\to \mathrm{\infty }}{a}^{(m)}(a)=\mathrm{\infty }$ and $\mu (m,a^{(m)}(a))=\mu (1,a)$ for $2\le m\le k$. So there exists $a_3\ge a_2$ such that $a^{(m)}(a)>a_2$ whenever $a\ge a_3$ and $2\le m\le k$. Let $a^{(1)}(a)=a$ and define

$${\underline{u}}_a(x,m)={\underline{u}}_{m,a^{(m)}(a)}(x),(x,m)\in M_k.$$

Then, by the definition of ${\underline{u}}_{m,a}$, when $a\ge a_3$, ${\underline{u}}_a\in C^0(M_k)$ is a locally admissible function satisfying

$$\begin{array}{c}{\underline{u}}_a(x,m)=\frac{\overline{R}}{2}{|x|}^2+C_m-O\left({|x|}^{2-n}\right),\text{when }|x|\to \mathrm{\infty },\hfill \\ {\underline{u}}_a(x,m)\le \frac{\overline{R}}{2}{|x|}^2+C_m,x\in {\mathbf{R}}^n,1\le m\le k,\hfill \\ \underset{x\to \overline{x}}{lim}{\underline{u}}_a(x,m)=-\mu (1,a),\overline{x}\in \partial \mathrm{\Sigma },1\le m\le k,\hfill \end{array}$$

and in the viscosity sense

$$F\left(\lambda \left(D^2{\underline{u}}_a\right)\right)\ge \sigma ,(x,m)\in M_k.$$

Step 2. We define the solution of (3.1) by the Perron method.

For $a\ge a_3$, let $S_a$ denote the set of admissible functions $V\in C^0(M_k)$ which can be extended to $\partial \mathrm{\Sigma }$ and satisfies

$$\begin{array}{c}F\left(\lambda \left(D^{2}V\right)\right)\ge \sigma ,(x,m)\in M_k,\hfill \\ \underset{x\to \overline{x}}{lim}V(x,m)\le -\mu (1,a),\overline{x}\in \mathrm{\Gamma },\hfill \\ V(x,m)\le \frac{\overline{R}}{2}{|x|}^2+C_m,x\in {\mathbf{R}}^n,1\le m\le k.\hfill \end{array}$$

It is obvious that ${\underline{u}}_a\in S_a$. Hence $S_a\ne \mathrm{\varnothing }$. Define

$$u_a(x,m)=sup\{V(x,m):V\in S_a\},(x,m)\in M_k.$$

Next we prove that $u_a$ is a viscosity solution of (3.1). From the definition of $u_a$, it is a viscosity subsolution of (3.1) and satisfies

$$u_a(x,m)\le \frac{\overline{R}}{2}{|x|}^2+C_m,x\in {\mathbf{R}}^n.$$

So we need only to prove that $u_a$ is a viscosity supersolution of (3.1) satisfying (3.2).

For any $x_0\in {\mathbf{R}}^n\setminus \partial \mathrm{\Sigma }$, fix $\epsilon >0$ such that $\overline{B}=\overline{B_{\epsilon }(x_0)}\subset {\mathbf{R}}^n\setminus \partial \mathrm{\Sigma }$. Then the lifting of $B$ into $M_k$ is the $k$ disjoint balls denoted as ${\{B^{(i)}\}}_{i=1}^k$. For any $(x,m)\in B^{(i)}$, by Lemma 2.3, there exists an admissible viscosity solution $\tilde{u}\in C^0(\overline{B^{(i)}})$ to the Dirichlet problem

$$\begin{array}{c}F\left(\lambda \left(D^2\tilde{u}\right)\right)=\sigma ,(x,m)\in B^{(i)},\hfill \\ \tilde{u}=u_a,(x,m)\in \partial B^{(i)}.\hfill \end{array}$$

By the comparison principle in [11],

$$u_a\le \tilde{u},(x,m)\in B^{(i)}.$$

(3.5)

Define

$$\psi (x,m)=\{\begin{array}{ll}\tilde{u}(x,m),& (x,m)\in B^{(i)},\\ u_a(x,m),& (x,m)\in M_k\setminus {\{B^{(i)}\}}_{i=1}^k.\end{array}$$

By Lemma 2.2,

$$F\left(\lambda (D^2\psi (x,m))\right)\ge \sigma ,x\in {\mathbf{R}}^n.$$

As

$$\begin{array}{c}F\left(\lambda \left(D^2\tilde{u}\right)\right)=\sigma =F\left(\lambda \left(D^{2}g\right)\right),(x,m)\in B^{(i)},\hfill \\ \tilde{u}=u_a\le g,(x,m)\in \partial B^{(i)},\hfill \end{array}$$

where $g(x,m)=\frac{\overline{R}}{2}{|x|}^2+C_m$, we have

$$\tilde{u}\le g,(x,m)\in \overline{B^{(i)}}$$

by the comparison principle in [11]. Therefore $\psi \in S_a$.

By the definition of $u_a,u_a\ge \psi$ in $M_k$. Consequently, $\tilde{u}\le u_a$ in $B^{(i)}$ and further $\tilde{u}=u_a$, $(x,m)\in B^{(i)}$ in view of (3.5). Since $x_0$ is arbitrary, we conclude that $u_a$ is an admissible viscosity solution of (3.1).

By the definition of $u_a$,

$${\underline{u}}_a\le u_a\le g,(x,m)\in M_k,$$

so $u_a$ satisfies (3.2) and we complete the proof of Theorem 3.1. □