# Global regularity results for a class of singular/degenerate fully nonlinear
elliptic equations

Sumiya Baasandorj  Department of Mathematical Sciences, Seoul National
University, Seoul 08826, Korea<EMAIL_ADDRESS>,  Sun-Sig Byun
Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 08826, Korea<EMAIL_ADDRESS>,  Ki-Ahm Lee
Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 08826, Korea<EMAIL_ADDRESS>and  Se-Chan
Lee  Research Institute of Mathematics, Seoul National University, Seoul
08826, Korea<EMAIL_ADDRESS>
###### Abstract.

We provide the Alexandroff-Bakelman-Pucci estimate and global $\displaystyle
C^{1,\alpha}$-regularity for a class of singular/degenerate fully nonlinear
elliptic equations. We also derive the existence of a viscosity solution to
the Dirichlet problem with the associated operator.

###### Key words and phrases:

Singular/degenerate fully nonlinear equations; Global regularity; Comparison
principle

###### 2010 Mathematics Subject Classification:

Primary 35B65; Secondary 35J60, 35J70, 35D40.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Alexandroff-Bakelman-Pucci estimate
  4. 4 Local Lipschitz estimates up to the boundary
  5. 5 Global $\displaystyle C^{1,\alpha}$-regularity
  6. 6 Comparison principle and Perron’s method

##  1\. Introduction

In this paper, we are concerned with the existence and global regularity
results for viscosity solutions of a class of singular/degenerate fully
nonlinear equations of the form

(1.3)
$\displaystyle\displaystyle\left\\{\begin{array}[]{rclcc}\Phi(x,|Du|)F(D^{2}u)&=&f(x)&\text{in
}&\Omega,\\\ u(x)&=&g(x)&\text{on }&\partial\Omega,\end{array}\right.$

where $\displaystyle F:\mathcal{S}(n)\rightarrow\mathbb{R}$ is a uniformly
$\displaystyle(\lambda,\Lambda)$-elliptic operator in the sense of (A1): ,
$\displaystyle\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty)$ is a
continuous map featuring degeneracy and singularity for the gradient described
as in (A2): , $\displaystyle f(\cdot)$ and $\displaystyle g(\cdot)$ are
suitable regular functions in the sense of (A3): , and $\displaystyle\Omega$
is a $\displaystyle C^{1,1}$-domain as in (A4): . We recall that, as a
consequence of Krylov-Safonov theory [9], viscosity solutions to the
homogeneous equation

$\displaystyle\displaystyle F(D^{2}u)=0\quad\text{in $\displaystyle
B_{1}$},\quad\text{where $\displaystyle F$ is uniformly
$\displaystyle(\lambda,\Lambda)$-elliptic,}$

belong to $\displaystyle C^{1,\overline{\alpha}}_{\mathrm{loc}}(B_{1})$ for a
universal constant
$\displaystyle\overline{\alpha}\equiv\overline{\alpha}(n,\lambda,\Lambda)\in(0,1)$.

Some special cases of (1.3), which are singular or degenerate PDEs in non-
divergence structure, have been widely studied in the past years. To be
precise, the local $\displaystyle C^{1,\alpha}$-regularity results for
degenerate fully nonlinear equations were developed in [2, 21] for
$\displaystyle\Phi(x,t)=t^{p}$ with $\displaystyle p\geq 0$, in [11, 16] for
$\displaystyle\Phi(x,t)=t^{p}+a(x)t^{q}$ with $\displaystyle 0\leq p\leq q$,
in [8] for $\displaystyle\Phi(x,t)=t^{p(x)}$ with $\displaystyle\inf
p(\cdot)>-1$, and in [17, 18] for
$\displaystyle\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}$ with $\displaystyle 0\leq
p(\cdot)\leq q(\cdot)$. On the other hand, comparison principle, Liouville
type results, and the ABP estimate are found mostly for
$\displaystyle\Phi(x,t)=t^{p}$ with $\displaystyle-1<p<0$; we refer to [5, 6,
14, 15, 20] for details. Finally, for both singular and degenerate general
operators which are considered in this paper, the local $\displaystyle
C^{1,\alpha}$-regularity with the optimality was shown by the authors [4].
Global counterpart of such local regularity results can be found in [7] for
$\displaystyle\Phi(x,t)=t^{p}$ with $\displaystyle p\geq 0$ and in [17] for
$\displaystyle\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}$ with $\displaystyle 0\leq
p(\cdot)\leq q(\cdot)$. We remark that global regularity of viscosity
solutions to (1.3) plays an essential role in proving the solvability of the
Dirichlet problem and in investigating obstacle problems; see [12, 13, 17].

The goal of this paper is to investigate the global regularity, involving the
ABP estimate and $\displaystyle C^{1,\alpha}$-estimate up to the boundary, for
both singular and degenerate fully nonlinear elliptic equations in a unified
way. Our first main theorem in this paper reads as follows:

######  Theorem 1.1 (Global $\displaystyle C^{1,\alpha}$-regularity).

Suppose the assumptions (A1): -(A4):  (to be described in the next section)
are in force. Let $\displaystyle\alpha$ be chosen to satisfy

(1.4)
$\displaystyle\displaystyle\alpha\in\begin{cases}(0,\overline{\alpha})\cap\left(0,\frac{1}{1+s(\Phi)}\right]\cap(0,\beta_{g})\
&\text{if $\displaystyle i(\Phi)\geq 0$},\\\
(0,\overline{\alpha})\cap\left(0,\frac{1}{1+s(\Phi)-i(\Phi)}\right]\cap(0,\beta_{g})\
&\text{if $\displaystyle-1<i(\Phi)<0$}.\end{cases}$

For any viscosity solution $\displaystyle u$ of

$\displaystyle\displaystyle\begin{cases}\Phi(x,|Du|)F(D^{2}u)=f\ &\text{in
$\displaystyle\Omega$},\\\ u=g\ &\text{on
$\displaystyle\partial\Omega$},\end{cases}$

there exists a constant $\displaystyle c\equiv
c(n,\lambda,\Lambda,i(\Phi),L,\alpha)$ such that $\displaystyle u\in
C^{1,\alpha}(\overline{\Omega})$ and

$\displaystyle\displaystyle\|u\|_{C^{1,\alpha}(\overline{\Omega})}\leq
c\left(1+\left\lVert
u\right\rVert_{L^{\infty}(\Omega)}+\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}+\left\lVert
f/\nu_{0}\right\rVert_{L^{\infty}(\Omega)}^{\frac{1}{1+i(\Phi)}}\right).$

The second main theorem concerning the solvability of the Dirichlet problem
follows from Theorem 1.1 together with Perron’s method.

######  Theorem 1.2 (Existence of viscosity solution).

Suppose the assumptions (A1): -(A4):  and (A5):  (to be stated in Section 6)
are in force. Then there exists a viscosity solution $\displaystyle u\in
C(\overline{\Omega})$ of (1.3).

Our strategy is to improve the global regularity of a viscosity solution
$\displaystyle u$ gradually. For this purpose, we begin with the ABP estimate
to show the global boundedness of solutions. Then, by constructing an
appropriate barrier function near the boundary, we capture the boundary
behavior of solutions in terms of a distance function. The comparison with a
distance function allows us to achieve a global Lipschitz estimate. In the
end, we prove the approximation lemma by employing the compactness argument
and then determine approximating linear functions in an iterative manner.

The main difficulty arises due to different behaviors of solutions relying on
the sign of $\displaystyle i(\Phi)$ defined in (A2): . To overcome such a
challenge, we first discuss the degenerate case ($\displaystyle i(\Phi)\geq
0$) in Section 4 and then transport the regular properties to a viscosity
solution of the singular case ($\displaystyle-1<i(\Phi)<0$) along with a
suitable modification of equations in Section 5. In addition, the degenerate
or singular character of PDEs leads to the lack of the comparison principle in
general settings. Therefore, we formulate special types of the comparison
principle: one is Lemma 4.2, where we exploit the smooth feature of barrier
functions, and the other is Lemma 6.2, in which we approximate the equations
to have a monotone property with respect to viscosity solution $\displaystyle
u$ of (1.3),

The paper is organized as follows. In Section 2, we present the assumptions
(A1): -(A4):  on the equation (1.3) and data to be used throughout the paper,
and then collect preliminary results related to our main theorem. Section 3 is
devoted to the proof of ABP estimate. The proofs for global $\displaystyle
C^{0,1}$-estimate and $\displaystyle C^{1,\alpha}$-estimate of viscosity
solutions $\displaystyle u$ of (1.3) are provided in Section 4 and Section 5,
respectively. Finally, in Section 6, we prove the comparison principle under
an additional assumption (A5):  to deduce the existence of a viscosity
solution by Perron’s method.

##  2\. Preliminaries

Throughout the paper, we denote by $\displaystyle
B_{r}(x_{0}):=\\{x\in\mathbb{R}^{n}:|x-x_{0}|<r\\}$ the open ball of
$\displaystyle\mathbb{R}^{n}$ with $\displaystyle n\geq 2$ centered at
$\displaystyle x_{0}$ with positive radius $\displaystyle r$. If the center is
clear in the context, we shall omit the center point by writing $\displaystyle
B_{r}\equiv B_{r}(x_{0})$. Moreover, $\displaystyle B_{1}\equiv
B_{1}(0)\subset\mathbb{R}^{n}$ denote the unit ball. We shall always denote by
$\displaystyle c$ a generic positive constant, possible varying line to line,
having dependencies on parameters using brackets, that is, for example
$\displaystyle c\equiv c(n,i(\Phi),\nu_{0})$ means that $\displaystyle c$
depends only on $\displaystyle n,i(\Phi)$ and $\displaystyle\nu_{0}$. For two
positive functions $\displaystyle f,g$, we write $\displaystyle f\lesssim g$
when there exists a universal constant $\displaystyle c>0$ such that
$\displaystyle f\leq cg$.

For a measurable map $\displaystyle g:\mathcal{B}\subset
B_{1}\rightarrow\mathbb{R}^{N}$ $\displaystyle(N\geq 1)$ with
$\displaystyle\beta\in(0,1]$ being a given number, we shall use the notation

$\displaystyle\displaystyle[g]_{C^{0,\beta}(\mathcal{B})}:=\sup\limits_{x\neq
y\in\mathcal{B}}\frac{|g(x)-g(y)|}{|x-y|^{\beta}},\quad[g]_{C^{0,\beta}}:=[g]_{C^{0,\beta}(B_{1})}.$

We now state the main assumptions in the paper.

(A1):

The operator $\displaystyle F:\mathcal{S}(n)\rightarrow\mathbb{R}$ is
continuous and uniformly $\displaystyle(\lambda,\Lambda)$-elliptic in the
sense that

$\displaystyle\displaystyle\lambda\text{tr}(N)\leq
F(M+N)-F(M)\leq\Lambda\text{tr}(N)$

holds with some constants $\displaystyle 0<\lambda\leq\Lambda$ and
$\displaystyle F(0)=0$, whenever $\displaystyle M,N\in\mathcal{S}(n)$ with
$\displaystyle N\geq 0$, where we denote by $\displaystyle\mathcal{S}(n)$ to
mean the set of $\displaystyle n\times n$ real symmetric matrices.

(A2):

$\displaystyle\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty)$ is a
continuous map satisfying the following properties:

  1. 1.:

There exist constants $\displaystyle s(\Phi)\geq i(\Phi)>-1$ such that the map
$\displaystyle\textstyle t\mapsto\frac{\Phi(x,t)}{t^{i(\Phi)}}$ is almost non-
decreasing with constant $\displaystyle L\geq 1$ in $\displaystyle(0,\infty)$
and the map $\displaystyle\textstyle t\mapsto\frac{\Phi(x,t)}{t^{s(\Phi)}}$ is
almost non-increasing with constant $\displaystyle L\geq 1$ in
$\displaystyle(0,\infty)$ for all $\displaystyle x\in\Omega$.

  2. 2.:

There exists constants $\displaystyle 0<\nu_{0}\leq\nu_{1}$ such that
$\displaystyle\displaystyle\nu_{0}\leq\Phi(x,1)\leq\nu_{1}$ for all
$\displaystyle x\in\Omega$.

(A3):

$\displaystyle f\in C(\Omega)\cap L^{\infty}({\Omega})$ and $\displaystyle
g\in C^{1,\beta_{g}}(\partial\Omega)$ for some
$\displaystyle\beta_{g}\in(0,1)$.

(A4):

$\displaystyle\Omega\subset\mathbb{R}^{n}$ is a bounded $\displaystyle
C^{1,1}$-domain.

The Pucci extremal operators $\displaystyle
P_{\lambda,\Lambda}^{\pm}:\mathcal{S}(n)\rightarrow\mathbb{R}$ are defined as

$\displaystyle\displaystyle
P_{\lambda,\Lambda}^{+}(M):=\Lambda\sum\limits_{\lambda_{k}>0}\lambda_{k}+\lambda\sum\limits_{\lambda_{k}<0}\lambda_{k}$

and

$\displaystyle\displaystyle
P_{\lambda,\Lambda}^{-}(M):=\lambda\sum\limits_{\lambda_{k}>0}\lambda_{k}+\Lambda\sum\limits_{\lambda_{k}<0}\lambda_{k},$

where $\displaystyle\\{\lambda_{k}\\}_{k=1}^{n}$ are the eigenvalues of the
matrix $\displaystyle M$. The $\displaystyle(\lambda,\Lambda)$-ellipticity of
the operator $\displaystyle F$ via the Pucci extremal operators can be
formulated as

$\displaystyle\displaystyle P_{\lambda,\Lambda}^{-}(N)\leq F(M+N)-F(M)\leq
P_{\lambda,\Lambda}^{+}(N)$

for all $\displaystyle M,N\in\mathcal{S}(n)$.

Before we proceed, we briefly explain several useful results concerning the
assumption (A4): . We may assume that $\displaystyle 0\in\partial\Omega$, and
there exist a ball $\displaystyle B=B_{R}(0)$ in $\displaystyle\mathbb{R}^{n}$
and $\displaystyle\phi\in C^{1,1}(\mathbb{R}^{n-1})$ such that
$\displaystyle\phi(0)=0,\nabla\phi(0)=0$ and

$\displaystyle\displaystyle\Omega\cap B\subset\\{y\in
B:y_{n}>\phi(y^{\prime})\\},\quad\partial\Omega\cap B=\\{y\in
B:y_{n}=\phi(y^{\prime})\\}.$

######  Definition 2.1 (The ball condition, [1]).

Let $\displaystyle\Omega$ be a bounded domain in
$\displaystyle\mathbb{R}^{n}$. We say that $\displaystyle D$ satisfies the
exterior ball condition (with radius $\displaystyle r$) if there exists
$\displaystyle r>0$ satisfying the following condition: for every
$\displaystyle x\in\partial\Omega$, there exists a point $\displaystyle
x^{e}\in\mathbb{R}^{n}\setminus\Omega$ such that $\displaystyle
B_{r}(x^{e})\subset\mathbb{R}^{n}\setminus\Omega$ and $\displaystyle
x\in\partial B_{r}(x^{e})$. Similarly, we can define the interior ball
condition. Finally, we say that $\displaystyle\Omega$ satisfies the ball
condition (with radius $\displaystyle r$) if $\displaystyle\Omega$ satisfies
both the exterior and the interior ball condition (with radius $\displaystyle
r$).

######  Lemma 2.2 ([1]).

Let $\displaystyle\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Then
$\displaystyle\Omega$ is a $\displaystyle C^{1,1}$-domain if and only if
$\displaystyle\Omega$ satisfies the ball condition.

On the other hand, for any vector $\displaystyle\xi\in\mathbb{R}^{n}$, we
consider a map $\displaystyle
G_{\xi}:\Omega\times\mathbb{R}^{n}\times\mathcal{S}(n)\rightarrow\mathbb{R}$
defined by

$\displaystyle\displaystyle G_{\xi}(x,p,M):=\Phi(x,|\xi+p|)F(M)-f(x)$

under the assumptions prescribed in (A1): -(A3): . In Section 4 and Section 5,
we shall focus on viscosity solutions of the equation

(2.1) $\displaystyle\displaystyle G_{\xi}(x,Du,D^{2}u)=0\text{ in }\Omega.$

We now give the definition of a viscosity solution $\displaystyle u$ of the
equation (2.1) as follows.

######  Definition 2.3.

A lower semicontinuous function $\displaystyle v$ is called a viscosity
supersolution of (2.1) if for all $\displaystyle x_{0}\in\Omega$ and
$\displaystyle\varphi\in C^{2}(\Omega)$ such that $\displaystyle v-\varphi$
has a local minimum at $\displaystyle x_{0}$, then

$\displaystyle\displaystyle
G_{\xi}(x_{0},D\varphi(x_{0}),D^{2}\varphi(x_{0}))\leq 0.$

An upper semicontinuous function $\displaystyle w$ is called is a viscosity
subsolution of (2.1) if for all $\displaystyle x_{0}\in\Omega$ and
$\displaystyle\varphi\in C^{2}(\Omega)$ such that $\displaystyle w-\varphi$
has a local maximum at $\displaystyle x_{0}$, there holds

$\displaystyle\displaystyle
G_{\xi}(x_{0},D\varphi(x_{0}),D^{2}\varphi(x_{0}))\geq 0.$

We say that $\displaystyle u\in C(\Omega)$ is a viscosity solution of (2.1) if
$\displaystyle u$ is a viscosity supersolution and a subsolution
simultaneously.

We also recall a concept of superjet and subjet introduced in [10].

######  Definition 2.4.

Let $\displaystyle v:\Omega\rightarrow\mathbb{R}$ be an upper semicontinuos
function and $\displaystyle w:\Omega\rightarrow\mathbb{R}$ be a lower
semicontinuous function. For every $\displaystyle x_{0}\in\Omega$, we define
the second order superjet of $\displaystyle v$ at $\displaystyle x_{0}$ by

$\begin{array}[]{rcl}J^{2,+}v(x_{0})&:=&\Bigg{\\{}(p,M)\in\mathbb{R}^{n}\times\mathcal{S}(n):v(x)\leq
v(x_{0})+\left\langle p,x-x_{0}\right\rangle\Bigg{.}\\\
&&\quad\left.+\frac{1}{2}\left\langle
M(x-x_{0}),x-x_{0}\right\rangle+o(|x-x_{0}|^{2})\text{ as }x\rightarrow
x_{0}\right\\}\end{array}$

and the second order subjet of $\displaystyle w$ at $\displaystyle x_{0}$ by

$\begin{array}[]{rcl}J^{2,-}w(x_{0})&:=&\Bigg{\\{}(p,M)\in\mathbb{R}^{n}\times\mathcal{S}(n):w(x)\geq
w(x_{0})+\left\langle p,x-x_{0}\right\rangle\Bigg{.}\\\
&&\quad\left.+\frac{1}{2}\left\langle
M(x-x_{0}),x-x_{0}\right\rangle+o(|x-x_{0}|^{2})\text{ as }x\rightarrow
x_{0}\right\\}.\end{array}$

  1. (i)

A couple $\displaystyle(p,M)\in\mathbb{R}^{n}\times\mathcal{S}(n)$ is a
limiting superjet of $\displaystyle v$ at $\displaystyle x_{0}\in\Omega$ if
there exists a sequence
$\displaystyle\\{x_{k},p_{k},M_{k}\\}\rightarrow\\{x,p,M\\}$ as $\displaystyle
k\rightarrow\infty$ in such a way that $\displaystyle(p_{k},M_{k})\in
J^{2,+}v(x_{k})$ and $\displaystyle\lim\limits_{k\to\infty}v(x_{k})=v(x_{0})$.

  2. (ii)

A couple $\displaystyle(p,M)\in\mathbb{R}^{n}\times\mathcal{S}(n)$ is a
limiting subjet of $\displaystyle w$ at $\displaystyle x\in B_{1}$ if there
exists a sequence $\displaystyle\\{x_{k},p_{k},M_{k}\\}\rightarrow\\{x,p,M\\}$
as $\displaystyle k\rightarrow\infty$ in such a way that
$\displaystyle(p_{k},M_{k})\in J^{2,-}w(x_{k})$ and
$\displaystyle\lim\limits_{k\to\infty}w(x_{k})=w(x_{0})$.

We finish this section by providing the interior $\displaystyle
C^{1,\alpha}$-regularity results shown in [4].

######  Theorem 2.5 ([4, Theorem 1.1]).

Let $\displaystyle u\in C(B_{1})$ be a viscosity solution of

$\displaystyle\displaystyle\Phi(x,|Du|)F(D^{2}u)=f(x)\quad\text{in}\quad
B_{1},$

under the assumptions (A1):  and (A2):  with $\displaystyle f\in
L^{\infty}(B_{1})$. Then $\displaystyle u\in
C_{\mathrm{loc}}^{1,\beta}(B_{1})$ for all $\displaystyle\beta>0$ satisfying

(2.2)
$\displaystyle\displaystyle\beta\in\begin{cases}(0,\overline{\alpha})\cap\left(\frac{1}{1+s(\Phi)}\right]\
&\text{if $\displaystyle i(\Phi)\geq 0$},\\\
(0,\overline{\alpha})\cap\left(\frac{1}{1+s(\Phi)-i(\Phi)}\right]\ &\text{if
$\displaystyle-1<i(\Phi)<0$}.\end{cases}$

Moreover, for every $\displaystyle\beta$ in (2.2), there exists a constant
$\displaystyle c\equiv c(n,\lambda,\Lambda,i(\Phi),L,\beta)$ such that

$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1/2})}+\sup_{x\neq y\in
B_{1/2}}\frac{|Du(x)-Du(y)|}{|x-y|^{\beta}}\leq
c\left(1+\|u\|_{L^{\infty}(B_{1})}+\|f/\nu_{0}\|_{L^{\infty}(B_{1})}^{\frac{1}{1+i(\Phi)}}\right).$

##  3\. Alexandroff-Bakelman-Pucci estimate

Before we develop $\displaystyle C^{0,1}$-regularity in Section 4 and
$\displaystyle C^{1,\alpha}$-regularity in Section 5, our study on the global
regularity starts with the Alexandroff-Bakelman-Pucci (ABP) estimate. In
short, the ABP estimate gives the supremum of $\displaystyle u$ over
$\displaystyle\Omega$ in terms of the supremum of $\displaystyle u$ on
$\displaystyle\partial\Omega$ and the $\displaystyle L^{n}$-norm of
$\displaystyle f$. In this section, we deduce an appropriate version of the
ABP estimate for a viscosity subsolution of (1.3). We refer to [14, 20, 22]
for similar results.

######  Definition 3.1.

For $\displaystyle v:\Omega\rightarrow\mathbb{R}$ and $\displaystyle R>0$, the
upper contact set is defined by

$\displaystyle\displaystyle\begin{split}&\Gamma^{+}(v,\Omega)=\left\\{x\in\Omega:\exists
p\in\mathbb{R}^{n}\text{ such that }u(y)\leq u(x)+\left\langle
p,y-x\right\rangle\text{ for all }y\in\Omega\right\\},\\\
&\Gamma^{+}_{R}(v,\Omega)=\left\\{x\in\Omega:\exists
p\in\overline{B_{R}(0)}\text{ such that }u(y)\leq u(x)+\left\langle
p,y-x\right\rangle\text{ for all }y\in\Omega\right\\}.\end{split}$

We are now ready to prove the ABP estimate.

######  Theorem 3.2 (Alexandroff-Bakelman-Pucci estimate).

Suppose that $\displaystyle u\in C(\overline{\Omega})$ is a viscosity
subsolution (resp. supersolution) of (1.3) in
$\displaystyle\\{x\in\Omega:u(x)>0\\}$ (resp.
$\displaystyle\\{x\in\Omega:u(x)<0\\}$) under the assumptions (A1): -(A2): .
Suppose that $\displaystyle f\in L^{n}(\Omega)\cap C(\Omega)$. Then, there
exists a constant $\displaystyle c\equiv
c(n,\lambda,i(\Phi),s(\Phi),L,\nu_{0})$ such that

(3.1)
$\displaystyle\displaystyle\sup\limits_{\Omega}u\leq\sup\limits_{\partial\Omega}g^{+}+c\operatorname{diam}(\Omega)\left(\max\left\\{\left\lVert
f^{-}\right\rVert^{\frac{1}{i(\Phi)+1}}_{L^{n}(\Gamma^{+}(u^{+}))},\left\lVert
f^{-}\right\rVert^{\frac{1}{s(\Phi)+1}}_{L^{n}(\Gamma^{+}(u^{+}))}\right\\}+1\right),$

respectively,

(3.2)
$\displaystyle\displaystyle\left(\sup\limits_{\Omega}u^{-}\leq\sup\limits_{\partial\Omega}g^{-}+c\operatorname{diam}(\Omega)\left(\max\left\\{\left\lVert
f^{+}\right\rVert^{\frac{1}{i(\Phi)+1}}_{L^{n}(\Gamma^{+}(u^{-}))},\left\lVert
f^{+}\right\rVert^{\frac{1}{s(\Phi)+1}}_{L^{n}(\Gamma^{+}(u^{-}))}\right\\}+1\right)\right).$

In particular, we have

(3.3) $\displaystyle\displaystyle\left\lVert
u\right\rVert_{L^{\infty}(\Omega)}\leq\left\lVert
g\right\rVert_{L^{\infty}(\partial\Omega)}+c\operatorname{diam}(\Omega)\left(\max\left\\{\left\lVert
f\right\rVert_{L^{n}(\Omega)}^{\frac{1}{i(\Phi)+1}},\left\lVert
f\right\rVert_{L^{n}(\Omega)}^{\frac{1}{s(\Phi)+1}}\right\\}+1\right)$

for some constant $\displaystyle c\equiv
c(n,\lambda,i(\Phi),s(\Phi),L,\nu_{0})>0$.

###### Proof.

The proof consists of two parts. In the first part, we prove the above theorem
for the viscosity subsolution $\displaystyle u\in C^{2}(\Omega)\cap
C(\overline{\Omega})$. In the second part, we consider $\displaystyle u\in
C(\overline{\Omega})$ via approximation based on the sup convolutions.

Part 1. Suppose that the subsolution $\displaystyle u$ belongs to
$\displaystyle C^{2}(\Omega)\cap C(\overline{\Omega})$. Let us define

$\displaystyle\displaystyle R_{0}\equiv
R_{0}(u):=\frac{1}{\operatorname{diam}(\Omega)}\left(\sup\limits_{x\in\Omega}u(x)-\sup\limits_{x\in\partial\Omega}u^{+}\right).$

The purpose is to obtain a certain estimate on $\displaystyle R_{0}$ in terms
of $\displaystyle\left\lVert f^{-}\right\rVert_{L^{n}(\Gamma^{+}(u^{+}))}$ and
$\displaystyle\operatorname{\textbf{data}}$, from which the estimate (3.1)
follows. Applying [22, Lemma 3.1], for all $\displaystyle R<R_{0}$, we find

(3.4)
$\displaystyle\displaystyle\int\limits_{B_{R}(0)}g(z)\,dz\leq\int\limits_{\Gamma_{R}^{+}(u^{+})}g(Du)\left|\det\left(D^{2}u\right)\right|\,dx\quad(\forall
g\in C(\mathbb{R}^{n}),\,g\geq 0)$

and

(3.5) $\displaystyle\displaystyle D^{2}u\leq 0\text{ on
}\Gamma_{R}^{+}(u^{+})\subset\\{x\in\Omega:u(x)>0\\}.$

Let us now discuss the behavior of $\displaystyle Du$ in the set
$\displaystyle\Gamma_{R}^{+}(u^{+})$. Let $\displaystyle
x_{0}\in\Gamma_{R}^{+}(u^{+})$ be any point. If $\displaystyle Du(x_{0})\neq
0$, then we are able to take $\displaystyle u$ as a test function in the
definition of viscosity subsolution. In turn, we have

$\displaystyle\displaystyle\Phi(x_{0},|Du(x_{0})|)F(D^{2}u(x_{0}))\geq
f(x_{0}).$

Then we see

$\displaystyle\displaystyle-f^{-}(x_{0})\leq
f(x_{0})\leq\Phi(x_{0},|Du(x_{0})|)F(D^{2}u(x_{0}))\leq\Phi(x_{0},|Du(x_{0})|)P_{\lambda,\Lambda}^{+}(D^{2}u(x_{0})).$

Recalling $\displaystyle D^{2}u(x_{0})\leq 0$ by (3.5), we find $\displaystyle
P_{\lambda,\Lambda}^{+}(D^{2}u(x_{0}))=\lambda\operatorname{tr}(D^{2}u(x_{0}))$
and

(3.6)
$\displaystyle\displaystyle\left(\frac{-\operatorname{tr}(D^{2}u(x_{0}))}{n}\right)^{n}\leq\left(\frac{f^{-}(x_{0})}{n\lambda\Phi(x_{0},|Du(x_{0})|)}\right)^{n}.$

If $\displaystyle Du(x_{0})=0$ and $\displaystyle D^{2}u(x_{0})\neq 0$, then
$\displaystyle x_{0}$ is a critical point of $\displaystyle u$. On the other
hand, recalling again (3.5), we have $\displaystyle D^{2}u(x_{0})<0$, which
means that $\displaystyle x_{0}$ is a non-degenerate critical point of
$\displaystyle u$. However, the set of non-degenerate critical points of
$\displaystyle u$ is countable since $\displaystyle u\in C^{2}(\Omega)$.

Let us recall also the following classical inequality,

$\displaystyle\displaystyle\det(A)\det(B)\leq\left(\frac{\operatorname{tr}(AB)}{n}\right)^{n}\text{
for all }A,B\in\mathcal{S}(n)\text{ with }A,B\geq 0.$

In turn, the last display together with (3.5) and (3.6) yields

(3.7) $\displaystyle\displaystyle\left|\det
D^{2}u(x)\right|\leq\left(\frac{f^{-}(x)}{n\lambda\Phi(x,|Du(x)|)}\right)^{n}$

for all $\displaystyle x\in\Gamma_{R}^{+}(u^{+})\setminus\mathcal{U}$, where
$\displaystyle\mathcal{U}=\\{x\in\Gamma_{R}^{+}(u^{+}):Du(x)=0\\}$. Now we
consider two steps depending on the sign of $\displaystyle i(\Phi)$.

Step 1. $\displaystyle i(\Phi)\geq 0$: Let us select $\displaystyle
g(z)=\min\left\\{|z|^{i(\Phi)n},|z|^{s(\Phi)n}\right\\}$ in (3.4). In turn,
recalling (3.7), we find

$\displaystyle\displaystyle I_{1}$
$\displaystyle\displaystyle:=\int\limits_{B_{R}(0)}\min\left\\{|z|^{i(\Phi)n},|z|^{s(\Phi)n}\right\\}\,dz$
$\displaystyle\displaystyle\leq\int\limits_{\Gamma_{R}^{+}(u^{+})\setminus\mathcal{U}}\min\left\\{|Du|^{i(\Phi)n},|Du|^{s(\Phi)n}\right\\}\left(\frac{f^{-}}{n\lambda\Phi(x,|Du|)}\right)^{n}\,dx$
$\displaystyle\displaystyle\leq\frac{L^{n}}{n^{n}\lambda^{n}\nu_{0}^{n}}\int\limits_{\Gamma_{R}^{+}(u^{+})}(f^{-})^{n}\,dx,$

where we have used (A2): . On the other hand, by co-area formula, we have

$\displaystyle\displaystyle I_{1}$
$\displaystyle\displaystyle=\int\limits_{0}^{R}\min\left\\{t^{i(\Phi)n},t^{s(\Phi)n}\right\\}\int\limits_{\partial
B_{t}(0)}\,dSdt=n\omega_{n}\int\limits_{0}^{R}\min\left\\{t^{i(\Phi)n},t^{s(\Phi)n}\right\\}t^{n-1}\,dt$
$\displaystyle\displaystyle=\left\\{\begin{array}[]{lr}\frac{\omega_{n}R^{(i(\Phi)+1)n}}{i(\Phi)+1}-\frac{\omega_{n}(s(\Phi)-i(\Phi))}{(i(\Phi)+1)(s(\Phi)+1)}&\text{if
}R\geq 1,\\\ \frac{\omega_{n}R^{(s(\Phi)+1)n}}{s(\Phi)+1}&\text{if
}R<1.\end{array}\right.$

Combining the last two displays, we arrive at (3.1).

Step 2: $\displaystyle-1<i(\Phi)<0$. In this case, we select $\displaystyle
g(z)=\left(\frac{|z|}{|z|+\delta}\right)^{-i(\Phi)n}\min\left\\{|z|^{i(\Phi)n},|z|^{s(\Phi)n}\right\\}$
for an arbitrary number $\displaystyle\delta>0$. Clearly, $\displaystyle g\in
C(\mathbb{R}^{n})$ and so we have

$\displaystyle\displaystyle I_{2}(\delta)$
$\displaystyle\displaystyle:=\int\limits_{B_{R}(0)}\left(\frac{|z|}{|z|+\delta}\right)^{-i(\Phi)n}\min\left\\{|z|^{i(\Phi)n},|z|^{s(\Phi)n}\right\\}\,dz$
$\displaystyle\displaystyle\leq\int\limits_{\Gamma_{R}^{+}(u^{+})\setminus\mathcal{U}}\left(\frac{|Du|}{|Du|+\delta}\right)^{-i(\Phi)n}\min\left\\{|Du|^{i(\Phi)n},|Du|^{s(\Phi)n}\right\\}\left(\frac{f^{-}}{n\lambda\Phi(x,|Du|)}\right)^{n}\,dx$
$\displaystyle\displaystyle\leq\frac{L^{n}}{n^{n}\lambda^{n}\nu_{0}^{n}}\int\limits_{\Gamma_{R}^{+}(u^{+})}(f^{-})^{n}\,dx,$

where we have used again (A2):  and the fact that $\displaystyle i(\Phi)<0$.
By using co-area formula and recalling that $\displaystyle-1<i(\Phi)<0$, we
get

$\displaystyle\displaystyle I_{2}(\delta)$
$\displaystyle\displaystyle=n\omega_{n}\int\limits_{0}^{R}\left(\frac{t}{t+\delta}\right)^{-i(\Phi)n}\min\left\\{t^{i(\Phi)n},t^{s(\Phi)n}\right\\}t^{n-1}\,dt.$

By applying Lebesgue’s dominated convergence theorem, we conclude

$\displaystyle\displaystyle\lim_{\delta\to 0^{+}}I_{2}(\delta)=$
$\displaystyle\displaystyle=n\omega_{n}\int\limits_{0}^{R}\min\left\\{t^{i(\Phi)n},t^{s(\Phi)n}\right\\}t^{n-1}\,dt$
$\displaystyle\displaystyle=\left\\{\begin{array}[]{lr}\frac{\omega_{n}R^{(i(\Phi)+1)n}}{i(\Phi)+1}-\frac{\omega_{n}(s(\Phi)-i(\Phi))}{(i(\Phi)+1)(s(\Phi)+1)}&\text{if
}R\geq 1,\\\ \frac{\omega_{n}R^{(s(\Phi)+1)n}}{s(\Phi)+1}&\text{if
}R<1.\end{array}\right.$

Combining the last two displays, we get (3.1).

Part 2. Let $\displaystyle u\in C(\overline{\Omega})$. Since we have ABP
estimates for $\displaystyle u\in C^{2}(\Omega)\cap C(\overline{\Omega})$, the
remainder of the proof can be argued similarly as in the proof of [22, Theorem
1.1], see also [14]. ∎

##  4\. Local Lipschitz estimates up to the boundary

By Theorem 3.2, any viscosity solution of (1.3) is bounded in $\displaystyle
L^{\infty}(\Omega)$ under the assumptions (A1): -(A3): . In this section, to
derive further Lipschitz estimates up to the boundary as in [5, 17], we
consider a bounded viscosity solution of

(4.3)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|Du|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ u(y)=g(y)&\text{on
$\displaystyle B\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

where the function $\displaystyle\phi$ is introduced in Section 2.

######  Remark 4.1 (Smallness regime).

Here we verify that, for a bounded viscosity solution $\displaystyle u$ of

(4.6)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|\xi+Du|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ u(y)=g(y)&\text{on
$\displaystyle B\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

we are able to assume

(4.7)
$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1,\,\|g\|_{C^{1,\beta_{g}}(B_{1}\cap\\{y_{n}=\phi(y^{\prime})\\})}\leq 1\text{
and
}\|f\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0},$

for some constant $\displaystyle 0<\varepsilon_{0}<1$ small enough, and also
$\displaystyle\nu_{0}=\nu_{1}=1$ in (A2): . In order to consider the problem
in a smallness regime as in (4.7), for a fixed ball $\displaystyle
B_{r}(x)\subset B$, we define
$\displaystyle\bar{u}:B_{1}\cap\\{y_{n}>\bar{\phi}(y^{\prime})\\}\rightarrow\mathbb{R}$
by

$\displaystyle\displaystyle\bar{u}(y):=\frac{u(ry+x)}{K}$

for a function $\displaystyle\bar{\phi}$ and positive constants $\displaystyle
K\geq 1\geq r$ to be determined later. It can be seen that
$\displaystyle\bar{u}$ is a viscosity solution of

(4.10)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\bar{\Phi}(y,|\bar{\xi}+D\bar{u}|)\bar{F}(D^{2}\bar{u})=\bar{f}(y)&\text{in
$\displaystyle B_{1}\cap\\{y_{n}>\bar{\phi}(y^{\prime})\\}$},\\\
\bar{u}(y)=\bar{g}(y)&\text{on $\displaystyle
B_{1}\cap\\{y_{n}=\bar{\phi}(y^{\prime})\\}$},\end{array}\right.$

where

$\displaystyle\displaystyle\displaystyle\bar{F}(M)$
$\displaystyle\displaystyle:=\frac{r^{2}}{K}F\left(\frac{K}{r^{2}}M\right),\quad\bar{\Phi}(y,t):=\frac{\Phi\left(ry+x,\frac{K}{r}t\right)}{\Phi\left(ry+x,\frac{K}{r}\right)},\quad\bar{f}(y):=\frac{r^{2}}{\Phi\left(ry+x,\frac{K}{r}\right)K}f(ry+x),$
$\displaystyle\displaystyle\bar{\xi}$
$\displaystyle\displaystyle:=\frac{r}{K}\xi,\quad\bar{\phi}(y^{\prime}):=\frac{\phi(ry^{\prime}+x^{\prime})-x_{n}}{r}\text{
and }\bar{g}(y):=\frac{g(ry+x)}{K}.$

Note that $\displaystyle\bar{F}$ is still a uniformly
$\displaystyle(\lambda,\Lambda)$-elliptic operator, the map
$\displaystyle\displaystyle t\mapsto\frac{\bar{\Phi}(y,t)}{t^{i(\Phi)}}$ is
almost non-decreasing and the map $\displaystyle\displaystyle
t\mapsto\frac{\bar{\Phi}(y,t)}{t^{s(\Phi)}}$ is almost non-increasing with the
same constants $\displaystyle L\geq 1$ and $\displaystyle s(\Phi)\geq
i(\Phi)>-1$ as in (A2): , and $\displaystyle\bar{\Phi}(y,1)=1$ for all
$\displaystyle y\in B_{1}$. It is immediate from the choice of $\displaystyle
r$ that $\displaystyle\|D^{2}\bar{\phi}\|_{\infty}\leq\|D^{2}\phi\|_{\infty}$
and

$\displaystyle\displaystyle\|\bar{g}\|_{C^{1,\beta_{g}}(B_{1}\cap\\{y_{n}=\bar{\phi}(y^{\prime})\\})}\leq\frac{1}{K}\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}.$

Moreover, the assumption (A2):  implies

$\displaystyle\displaystyle\left\lVert\bar{f}\right\rVert_{L^{\infty}(B_{1}\cap\\{y_{n}>\bar{\phi}(y^{\prime})\\})}\leq\frac{Lr^{2+i(\Phi)}}{\nu_{0}K^{1+i(\Phi)}}\left\lVert
f\right\rVert_{L^{\infty}(\Omega)}.$

By recalling $\displaystyle i(\Phi)>-1$ and setting

$\displaystyle\displaystyle K:=2\left(1+\left\lVert
u\right\rVert_{L^{\infty}(\Omega)}+\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}+\left[\frac{L}{\nu_{0}}\left\lVert
f\right\rVert_{L^{\infty}(\Omega)}\right]^{\frac{1}{1+i(\Phi)}}\right)$

and

$\displaystyle\displaystyle r:=\varepsilon_{0}^{\frac{1}{2+i(\Phi)}},$

we see that $\displaystyle\bar{u}$ solves the equation (4.10) under the
smallness regime in (4.7).

If we have special conditions on a viscosity supersolution (or subsolution),
then we can apply the comparison principle without an additional structure
condition such as (A5):  in Section 6. See Section 6 for more comments on the
comparison principle.

######  Lemma 4.2 (Comparison principle I).

Let $\displaystyle f_{1},f_{2}\in C(\overline{\Omega})$ with $\displaystyle
f_{1}>f_{2}$ and $\displaystyle v\in C(\overline{\Omega})$ be a viscosity
subsolution of $\displaystyle\Phi(y,|Du|)F(D^{2}u)=f_{1}(y)$ in
$\displaystyle\Omega$. Moreover, let $\displaystyle w\in
C(\overline{\Omega})\cap C^{2}(\Omega)$ be a viscosity supersolution of
$\displaystyle\Phi(y,|Du|)F(D^{2}u)=f_{2}(y)$. If $\displaystyle v\leq w$ on
$\displaystyle\partial\Omega$, then $\displaystyle v\leq w$ in
$\displaystyle\Omega$.

###### Proof.

By contradiction, we suppose that

$\displaystyle\displaystyle\max_{x\in\overline{\Omega}}(v(x)-w(x))>0$

and the maximum is achieved at a point $\displaystyle\hat{x}\in\Omega$. Since
$\displaystyle v$ is a viscosity subsolution, $\displaystyle w\in C^{2}$ and
$\displaystyle v-w$ has a local maximum at $\displaystyle\hat{x}$, the
definition of viscosity subsolutions yields

$\displaystyle\displaystyle\Phi(\hat{x},|Dw(\hat{x})|)F(D^{2}w(\hat{x}))\geq
f_{1}(\hat{x}).$

On the other hand, since $\displaystyle w$ is a viscosity supersolution, we
have

$\displaystyle\displaystyle\Phi(\hat{x},|Dw(\hat{x})|)F(D^{2}w(\hat{x}))\leq
f_{2}(\hat{x}),$

which leads to the contradiction. ∎

######  Lemma 4.3.

Let $\displaystyle g\in C^{1,\beta_{g}}(\partial\Omega)$. Let $\displaystyle
d$ be the distance to the hypersurface
$\displaystyle\\{y_{n}=\phi(y^{\prime})\\}$.
Then for every $\displaystyle r\in(0,1)$ and $\displaystyle\gamma\in(0,1)$,
there exists $\displaystyle\delta_{0}>0$ depending on
$\displaystyle\|f\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}$,
$\displaystyle\lambda$, $\displaystyle\Lambda$, $\displaystyle\Omega$,
$\displaystyle r$, $\displaystyle L$, $\displaystyle\nu_{0}$ and
$\displaystyle\mathrm{Lip}_{g}(\partial\Omega)$ such that for every
$\displaystyle 0<\delta<\delta_{0}$, if $\displaystyle u$ is a viscosity
solution of (4.3) with
$\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1$, then

$\displaystyle\displaystyle|u(y^{\prime},y_{n})-g(y^{\prime})|\leq\frac{6}{\delta}\frac{d(y)}{1+d(y)^{\gamma}}\quad\text{in
$\displaystyle B_{r}(0)\cap\\{y_{n}>\phi(y^{\prime})\\}$.}$

###### Proof.

We separate two cases: (i) $\displaystyle g\equiv 0$, (ii) $\displaystyle g$
is not identically zero.

  1. (i)

($\displaystyle g\equiv 0$) In this case, we have
$\displaystyle\|u\|_{L^{\infty}}\leq 1$ and so we will only consider the
smaller set $\displaystyle\Omega_{\delta}:=\\{y\in\Omega:d(y)<\delta\\}$.
Moreover, we choose $\displaystyle\delta_{1}>0$ such that if $\displaystyle
d(y)<\delta_{1}$, then $\displaystyle d$ belongs to $\displaystyle C^{2}$ and
$\displaystyle|D^{2}d|\leq K$ for some universal constant $\displaystyle K>0$.

The proof relies on the construction of upper and lower barriers. For this
purpose, we define a function $\displaystyle w\in C^{2}(\Omega_{\delta})$ by

$\displaystyle\displaystyle
w(y)=\left\\{\begin{array}[]{ll}\frac{2}{\delta}\frac{d(y)}{1+d^{\gamma}(y)}&\text{for
$\displaystyle|y|<r$},\\\
\frac{2}{\delta}\frac{d(y)}{1+d^{\gamma}(y)}+\frac{1}{(1-r)^{3}}(|y|-r)^{3}&\text{for
$\displaystyle|y|\geq r$}.\end{array}\right.$

By following the argument in [7, Lemma 2.2], we have $\displaystyle w\geq u$
on
$\displaystyle\partial(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\}\cap\Omega_{\delta})$.
Moreover, it is easily checked that $\displaystyle|Dw|\geq\frac{1}{4\delta}$
when $\displaystyle\delta\leq\frac{1-r}{12}$ and so if we choose
$\displaystyle\delta<1/4$, then $\displaystyle|Dw|\geq 1$. Moreover, we can
calculate

$\displaystyle\displaystyle\mathcal{P}_{\lambda,\Lambda}^{+}(D^{2}w)\leq-2\gamma\delta^{\gamma-2}\lambda\frac{1+\gamma}{(1+\delta^{\gamma})^{3}}+\frac{2}{\delta}nK\Lambda+\frac{6n\Lambda}{(1-r)^{2}}\lesssim-\delta^{\gamma-2}+\delta^{-1}.$

Since $\displaystyle\gamma-2<-1<0$, we can further choose
$\displaystyle\delta\in(0,1)$ small enough so that
$\displaystyle\mathcal{P}_{\lambda,\Lambda}^{+}(D^{2}w)<0$. Then, by recalling
(A2): ,

$\displaystyle\displaystyle\Phi(x,|Dw|)\mathcal{P}_{\lambda,\Lambda}^{+}(D^{2}w)\lesssim-L\nu_{0}|Dw|^{i(\Phi)}(\delta^{\gamma-2}-\delta^{-1})\leq-L\nu_{0}(\delta^{\gamma-i(\Phi)-2}-\delta^{-1-i(\Phi)}).$

Since $\displaystyle\gamma-i(\Phi)-2<-1-i(\Phi)<0$, we finally choose
$\displaystyle\delta\in(0,1)$ small enough so that

$\displaystyle\displaystyle\Phi(x,|Dw|)\mathcal{P}_{\lambda,\Lambda}^{+}(D^{2}w)<-\|f\|_{\infty}-1.$

By applying Lemma 4.2, we conclude that

$\displaystyle\displaystyle u\leq
w=\frac{2}{\delta}\frac{d(y)}{1+d^{\gamma}(y)}\quad\text{in $\displaystyle
B_{r}(0)\cap\\{y_{n}>\phi(y^{\prime})\\}$.}$

The lower bound for $\displaystyle u$ can be obtained in a similar argument.

  2. (ii)

($\displaystyle g$ is not identically zero) This case follows from the same
argument as in [7, Lemma 2.2].

∎

The main theorem in this section is the following boundary Lipschitz estimate,
whose proof relies on Lemma 4.3 and the Ishii-Lions Lemma [10].

######  Theorem 4.4 (Lipschitz estimates for $\displaystyle\xi=0$).

Let $\displaystyle g$ be a Lipschitz continuous function. Suppose that
$\displaystyle u$ satisfies (4.3) with
$\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1$. Then for every $\displaystyle r\in(0,1)$, we have $\displaystyle u\in
C^{0,1}(B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\})$ and

(4.11)
$\displaystyle\displaystyle\|u\|_{C^{0,1}(B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
C(n,\lambda,\Lambda,i(\Phi),s(\Phi),r,L,\mathrm{Lip}_{g}(\partial\Omega),\|f\|_{L^{\infty}(\Omega)}).$

###### Proof.

Let $\displaystyle r_{1}\in(r,1)$ be fixed. For $\displaystyle x_{0}\in
B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\}$, we define

$\displaystyle\displaystyle\Psi(x,y)\coloneqq
u(x)-u(y)-M\omega(|x-y|)-L\left(|x-x_{0}|^{2}+|y-x_{0}|^{2}\right),$

where

$\displaystyle\displaystyle\omega(s):=\begin{cases}s-\omega_{0}s^{\frac{3}{2}}\
&\text{if $\displaystyle s\leq
s_{0}:=\left(\frac{2}{3\omega_{0}}\right)^{2}$},\\\ \omega(s_{0})\ &\text{if
$\displaystyle s\geq s_{0}$.}\end{cases}$

We claim that for $\displaystyle L,M\gg 1$ large enough,

(4.12) $\displaystyle\displaystyle\Psi(x,y)\leq 0\quad\text{for all
$\displaystyle(x,y)\in(B_{r_{1}}\cap\overline{\Omega})\times(B_{r_{1}}\cap\overline{\Omega})$}.$

Note that this inequality implies the desired Lipschitz estimate.

First of all, suppose that $\displaystyle y\in
B_{r_{1}}\cap\\{y_{n}=\phi(y^{\prime})\\}$. Then by Lemma 4.3, there exists a
constant $\displaystyle K_{0}>0$ such that

$\displaystyle\displaystyle|u(z)-g(z^{\prime})|\leq
K_{0}d(z,\partial\Omega)\quad\text{for $\displaystyle z\in
B_{r_{1}}\cap\\{y_{n}>\phi(y^{\prime})\\}$},$

which implies that

$\displaystyle\displaystyle|u(x)-u(y)|$
$\displaystyle\displaystyle\leq|u(x^{\prime},x_{n})-u(x^{\prime},\phi(x^{\prime}))|+|u(x^{\prime},\phi(x^{\prime}))-u(y^{\prime},\phi(y^{\prime}))|$
$\displaystyle\displaystyle\leq
K_{0}d(x,\partial\Omega)+\mathrm{Lip}_{g}(\partial\Omega)|x^{\prime}-y^{\prime}|\leq(K_{0}+\mathrm{Lip}_{g}(\partial\Omega))|x-y|.$

Therefore, if we choose $\displaystyle M/3\geq
K_{0}+\mathrm{Lip}_{g}(\partial\Omega)$, then

$\displaystyle\displaystyle\Psi(x,y)\leq
M\left(\frac{|x-y|}{3}-\omega(|x-y|)\right)-L\left(|x-x_{0}|^{2}+|y-x_{0}|^{2}\right)\leq
0.$

We now prove (4.12) by contradiction; suppose that there exists some point
$\displaystyle(\hat{x},\hat{y})\in(B_{r_{1}}\cap\overline{\Omega})\times(B_{r_{1}}\cap\overline{\Omega})$
such that

$\displaystyle\displaystyle\Psi(\hat{x},\hat{y})=\max_{(B_{r_{1}}\cap\overline{\Omega})\times(B_{r_{1}}\cap\overline{\Omega})}\Psi(x,y)>0.$

Here, we also choose $\displaystyle
L>\max\left\\{\frac{8}{(r_{1}-r)^{2}},\frac{1}{2(r+r_{1})}\right\\}$. Then we
can easily check that (i) $\displaystyle\hat{x}\neq\hat{y}$; (ii)
$\displaystyle\hat{x},\hat{y}\in B_{r_{1}}\cap\\{y_{n}>\phi(y^{\prime})\\}$;
(iii) $\displaystyle\hat{x},\hat{y}\in B_{\frac{r_{1}+r}{2}}$. Thus, by
applying Ishii-Lions Lemma [10, Theorem 3.2], we see that, for every
$\displaystyle\varepsilon>0$ sufficiently small, there exist $\displaystyle
X,Y\in\mathcal{S}^{n}$ such that

$\displaystyle\displaystyle(M\omega^{\prime}(|\hat{x}-\hat{y}|)\hat{a}+2L(\hat{x}-x_{0}),X)\in\overline{J}^{2,+}u(\hat{x}),$
$\displaystyle\displaystyle(M\omega^{\prime}(|\hat{x}-\hat{y}|)\hat{a}-2L(\hat{y}-x_{0}),-Y)\in\overline{J}^{2,-}u(\hat{y}),$
(4.13) $\displaystyle\displaystyle\begin{pmatrix}X&0\\\ 0&Y\end{pmatrix}\leq
M\begin{pmatrix}Z&-Z\\\
-Z&Z\end{pmatrix}+(2L+\varepsilon)\begin{pmatrix}I&0\\\ 0&I\end{pmatrix},$

where

$\displaystyle\displaystyle
Z=\omega^{\prime\prime}(|\hat{x}-\hat{y}|)\hat{a}\otimes\hat{a}+\frac{\omega^{\prime}(|\hat{x}-\hat{y}|)}{|\hat{x}-\hat{y}|}(I-\hat{a}\otimes\hat{a})\quad\text{for
$\displaystyle\hat{a}\coloneqq\frac{\hat{x}-\hat{y}}{|\hat{x}-\hat{y}|}$.}$

For simplicity, we write $\displaystyle q_{x}\coloneqq
M\omega^{\prime}(|\hat{x}-\hat{y}|)\hat{a}+2L(\hat{x}-x_{0})$ and
$\displaystyle q_{y}\coloneqq
M\omega^{\prime}(|\hat{x}-\hat{y}|)\hat{a}-2L(\hat{y}-x_{0})$. We first choose
$\displaystyle\omega_{0}>0$ small enough so that $\displaystyle s_{0}\geq
2>r+r_{1}$. Note that $\displaystyle t\mapsto\omega^{\prime}(t)$ is decreasing
on $\displaystyle t\in[0,s_{0}]$. If we choose $\displaystyle M>0$ large
enough so that $\displaystyle
2L(r+r_{1})\leq\frac{M}{2}\omega^{\prime}(r+r_{1})$, then we have
$\displaystyle
2L|\hat{x}-x_{0}|,2L|\hat{y}-x_{0}|\leq\frac{M}{2}\omega^{\prime}(|\hat{x}-\hat{y}|)$.
In particular, we obtain

(4.14)
$\displaystyle\displaystyle\frac{M}{2}\omega^{\prime}(|\hat{x}-\hat{y}|)\leq|q_{x}|,|q_{y}|\leq{2M}\omega^{\prime}(|\hat{x}-\hat{y}|),$

and by the choice of $\displaystyle L$, we also know that
$\displaystyle|q_{x}|,|q_{y}|\geq 1$.

On the other hand, for $\displaystyle X$ and $\displaystyle Y$, we will use
the matrix inequality (4). First, by evaluating a vector of the form
$\displaystyle(\xi,\xi)$ for any $\displaystyle\xi\in\mathbb{R}^{n}$, we have

$\displaystyle\displaystyle(X+Y)\xi\cdot\xi\leq 6L|\xi|^{2},$

which implies that any eigenvalues of $\displaystyle X+Y$ are less than
$\displaystyle 6L$. Moreover, by applying the matrix inequality (4) for
$\displaystyle(\hat{a},-\hat{a})$, we observe

$\displaystyle\displaystyle(X+Y)\hat{a}\cdot\hat{a}\leq
4M\omega^{\prime\prime}(|\hat{x}-\hat{y}|)+6L=-3M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6L.$

In other words, at least one eigenvalue of $\displaystyle X+Y$ is less than
$\displaystyle-3M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6L$. Therefore, by the
definition of the Pucci operator, we have

$\displaystyle\displaystyle\mathcal{P}_{\lambda,\Lambda}^{+}(X+Y)$
$\displaystyle\displaystyle\leq\lambda(-3M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6L)+6\Lambda(n-1)L$
$\displaystyle\displaystyle=-3\lambda
M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6[\Lambda(n-1)+\lambda]L.$

We now employ the definition of limiting superjet and limiting subjet:

$\displaystyle\displaystyle\Phi(\hat{x},|q_{x}|)F(X)\geq
f(\hat{x})\geq-\|f\|_{\infty},$
$\displaystyle\displaystyle\Phi(\hat{y},|q_{y}|)F(-Y)\leq
f(\hat{y})\leq\|f\|_{\infty}.$

Since $\displaystyle|q_{x}|,|q_{y}|\geq 1$, an application of (A2):  and
(4.14) yields that

$\displaystyle\displaystyle\Phi(\hat{x},|q_{x}|)\geq
L\nu_{0}|q_{x}|^{i(\Phi)},\quad\Phi(\hat{y},|q_{y}|)\geq
L\nu_{0}|q_{y}|^{i(\Phi)}.$

Moreover, (A1):  shows that

$\displaystyle\displaystyle
F(X)-F(-Y)\leq\mathcal{P}^{+}_{\lambda,\Lambda}(X+Y)\leq-3\lambda
M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6[\Lambda(n-1)+\lambda]L.$

Combining these results, we have

$\displaystyle\displaystyle-\|f\|_{\infty}\left(|q_{x}|^{-i(\Phi)}+|q_{y}|^{-i(\Phi)}\right)\leq-3\lambda
M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6[\Lambda(n-1)+\lambda]L.$

We now split into two cases depending on the sign of $\displaystyle i(\Phi)$:

  1. (i)

($\displaystyle i(\Phi)\geq 0$) Since $\displaystyle|q_{x}|,|q_{y}|\geq 1$ and
$\displaystyle|\hat{x}-\hat{y}|\leq 1$, we conclude that

$\displaystyle\displaystyle 3\lambda M\omega_{0}\leq
2\|f\|_{\infty}+6[\Lambda(n-1)+\lambda]L,$

which does not hold for sufficiently large $\displaystyle M>0$.

  2. (ii)

($\displaystyle-1<i(\Phi)<0$) Recalling that
$\displaystyle|q_{x}|,|q_{y}|\leq{2M}\omega^{\prime}(|\hat{x}-\hat{y}|)\leq
2M$, we derive

$\displaystyle\displaystyle 3\lambda M\omega_{0}\leq
2\|f\|_{\infty}(2M)^{-i(\Phi)}+6[\Lambda(n-1)+\lambda]L.$

Since $\displaystyle-i(\Phi)<1$, this inequality does not hold for
sufficiently large $\displaystyle M>0$.

This finishes the proof for (4.12). ∎

On the other hand, for a modified equation (2.1), we can prove the boundary
Lipschitz esimate, provided that $\displaystyle|\xi|$ is large. In short, the
boundary Lipschitz estimates hold when either

  1. (i)

$\displaystyle\xi=0$ with $\displaystyle i(\Phi)>-1$ (Lemma 4.3 and Theorem
4.4) or

  2. (ii)

$\displaystyle|\xi|$ is large with $\displaystyle i(\Phi)\geq 0$ (Lemma 4.5
and Theorem 4.6).

######  Lemma 4.5.

Let $\displaystyle g$ be Lipschitz continuous on $\displaystyle\partial\Omega$
and $\displaystyle\xi\in\mathbb{R}^{n}$ with $\displaystyle|\xi|=1$. Then for
every $\displaystyle r\in(0,1)$ and $\displaystyle\gamma\in(0,1)$, there
exists $\displaystyle\delta>0$ depending on $\displaystyle\lambda$,
$\displaystyle\Lambda$, $\displaystyle s(\Phi)$, $\displaystyle r$ and
$\displaystyle\mathrm{Lip}_{g}(\partial\Omega)$ such that for $\displaystyle
0\leq b<\frac{\delta}{6}$, any viscosity solution $\displaystyle u$ of

(4.17)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|\xi+bDu|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ u(y)=g(y)&\text{on
$\displaystyle B_{1}\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

with

$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1\quad\text{and}\quad\|f\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0}$

satisfies

$\displaystyle\displaystyle|u(y^{\prime},y_{n})-g(y^{\prime})|\leq\frac{6}{\delta}\frac{d(y)}{1+d(y)^{\gamma}}\quad\text{in
$\displaystyle B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\}$.}$

###### Proof.

As in the proof of Lemma 4.3, we may suppose that $\displaystyle g\equiv 0$
and construct a barrier function in a local domain
$\displaystyle\Omega_{\delta}:=\\{y\in\Omega:d(y)<\delta\\}$. If
$\displaystyle b=0$, then there is no degeneracy with respect to the gradient
$\displaystyle Du$ and so the result holds. Thus, we also suppose that
$\displaystyle b>0$.

We now define a function $\displaystyle w\in C^{2}(\Omega_{\delta})$ by

$\displaystyle\displaystyle
w(y)=\left\\{\begin{array}[]{ll}\frac{2}{\delta}\frac{d(y)}{1+d^{\gamma}(y)}&\text{for
$\displaystyle|y|<r$},\\\
\frac{2}{\delta}\frac{d(y)}{1+d^{\gamma}(y)}+\frac{1}{(1-r)^{3}}(|y|-r)^{3}&\text{for
$\displaystyle|y|\geq r$}.\end{array}\right.$

We recall that $\displaystyle w\geq u$ on
$\displaystyle\partial(B\cap\\{y_{n}>\phi(y^{\prime})\\}\cap\Omega_{\delta})$
and

$\displaystyle\displaystyle\mathcal{P}_{\lambda,\Lambda}^{+}(D^{2}w)\leq-2\gamma\delta^{\gamma-2}\lambda\frac{1+\gamma}{(1+\delta^{\gamma})^{3}}+\frac{2}{\delta}nK\Lambda+\frac{6n\Lambda}{(1-r)^{2}}\lesssim-\delta^{\gamma-2}+\delta^{-1}.$

On the other hand, since

$\displaystyle\displaystyle
Dw(y)=\left\\{\begin{array}[]{ll}\frac{2}{\delta}\frac{1+(1-\gamma)d^{\gamma}}{(1+d^{\gamma})^{2}}Dd&\text{for
$\displaystyle|y|<r$},\\\
\frac{2}{\delta}\frac{1+(1-\gamma)d^{\gamma}}{(1+d^{\gamma})^{2}}Dd+\frac{y}{|y|}\frac{3}{(1-r)^{3}}(|y|-r)^{2}&\text{for
$\displaystyle|y|\geq r$},\end{array}\right.$

we have $\displaystyle|Dw|\leq\frac{3}{\delta}$ provided that
$\displaystyle\delta\leq\frac{1-r}{3}$. As a consequence, we derive

$\displaystyle\displaystyle\frac{1}{2}\leq|\xi+bDw|\leq\frac{3}{2}\quad\text{for
$\displaystyle 0<b<\frac{\delta}{6}$},$

and so we conclude that

$\displaystyle\displaystyle\Phi(y,|\xi+bDw|)F(D^{2}w)<-\|f\|_{\infty}-1,\quad\text{for
sufficiently small $\displaystyle\delta>0$.}$

Lemma 4.2 yields the upper bound for $\displaystyle u$, and the remaining part
can be done as in Lemma 4.3. ∎

Note that Lemma 4.5 holds for any $\displaystyle i(\Phi)>-1$, while Theorem
4.6 holds only for the degenerate case, $\displaystyle i(\Phi)\geq 0$.

######  Theorem 4.6 (Lipschitz estimates for large $\displaystyle|\xi|$ with
$\displaystyle i(\Phi)\geq 0$).

Let $\displaystyle g$ be Lipschitz continuous on $\displaystyle\partial\Omega$
and $\displaystyle\xi\in\mathbb{R}^{n}$. Assume that $\displaystyle u$ is a
viscosity solution of

(4.20)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|\xi+Du|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ u(y)=g(y)&\text{on
$\displaystyle B_{1}\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

with

$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1\quad\text{and}\quad\|f\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0}.$

Then for all $\displaystyle r\in(0,1)$, there exists $\displaystyle
p_{0}=p_{0}(\lambda,\Lambda,n,i(\Phi),s(\Phi),r,\varepsilon_{0},\mathrm{Lip}_{g}(\partial\Omega))>0$,
such that if $\displaystyle|\xi|>p_{0}$, then $\displaystyle u\in
C^{0,1}(B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\})$ and we have the estimate

$\displaystyle\displaystyle\|u\|_{C^{0,1}(B_{r}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
C(\lambda,\Lambda,n,i(\Phi),s(\Phi),r,\varepsilon_{0},\mathrm{Lip}_{g}(\partial\Omega)).$

###### Proof.

Since the proof is similar to the one of Theorem 4.4, here we concentrate on
the differences.

  1. (i)

We first need to show that if $\displaystyle x$ or $\displaystyle y$ belongs
to $\displaystyle B_{r_{1}}\cap\\{y_{n}=\phi(y^{\prime})\\}$, then
$\displaystyle\Psi(x,y)\leq 0$. In the case of Theorem 4.4, this result
immediately followed from Lemma 4.3. In a similar manner, it is enough to
apply Lemma 4.5 for a solution $\displaystyle u$. More precisely, if
$\displaystyle u$ is a solution of (4.20), then $\displaystyle u$ solves

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\tilde{\Phi}\left(y,\left|{\xi}/{|\xi|}+bDu\right|\right)F(D^{2}u)=\tilde{f}(y)&\text{in
$\displaystyle B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ u(y)=g(y)&\text{on
$\displaystyle B_{1}\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

where
$\displaystyle\tilde{\Phi}(y,t):=\frac{\Phi(y,|\xi|t)}{\Phi(y,|\xi|)},\tilde{f}(y):=\frac{f(y)}{\Phi(y,|\xi|)}$
and $\displaystyle b:=1/|\xi|$. Thus, if we choose $\displaystyle
p_{0}>\max\\{1,6/\delta\\}$, then we can apply Lemma 4.5 for $\displaystyle
u$.

  2. (ii)

We next follow the contradiction argument of Theorem 4.4 and the difference
occurs when we employ the definition of limiting superjet and subjet:

$\displaystyle\displaystyle\Phi(\hat{x},|\xi+q_{x}|)F(X)\geq-\|f\|_{\infty},$
$\displaystyle\displaystyle\Phi(\hat{y},|\xi+q_{y}|)F(-Y)\leq\|f\|_{\infty}.$

This is due to the difference between equations (4.3) and (4.20), but we are
still able to derive a contradiction. Recalling that
$\displaystyle|q_{x}|,|q_{y}|\leq 2M\omega^{\prime}(|\hat{x}-\hat{y}|)\leq
2M$, if we choose $\displaystyle p_{0}>3M$, then we have

$\displaystyle\displaystyle|\xi+q_{x}|,|\xi+q_{y}|\geq M.$

Combining this estimate with

$\displaystyle\displaystyle-\|f\|_{\infty}\left(|\xi+q_{x}|^{-i(\Phi)}+|\xi+q_{y}|^{-i(\Phi)}\right)\leq-3\lambda
M\omega_{0}|\hat{x}-\hat{y}|^{-1/2}+6[\Lambda(n-1)+\lambda]L,$

we conclude that

$\displaystyle\displaystyle 3\lambda M\omega_{0}\leq
2\|f\|_{\infty}M^{-i(\Phi)}+6[\Lambda(n-1)+\lambda]L,$

which is a contradiction. In this step, we have exploited the condition
$\displaystyle i(\Phi)\geq 0$.

∎

##  5\. Global $\displaystyle C^{1,\alpha}$-regularity

We start this section with several reductions of the proof of Theorem 1.1.
First of all, by recalling Theorem 2.5 which provides the interior
$\displaystyle C^{1,\alpha}$-estimate of viscosity solutions, it is enough to
show the local $\displaystyle C^{1,\alpha}$-estimate up to the boundary. Next,
by following the proof of [4, Theorem 1.1], we shall consider the degenerate
case ($\displaystyle i(\Phi)\geq 0$) first, and then utilize this result for
the singular case ($\displaystyle-1<i(\Phi)<0$).

######  Lemma 5.1 ($\displaystyle C^{1,\alpha}$-regularity up to the boundary;
degenerate case).

Suppose the assumptions (A1): -(A4):  are in force with $\displaystyle
i(\Phi)\geq 0$, $\displaystyle\phi(0)=0,\nabla\phi(0)=0$. Let
$\displaystyle\alpha$ be chosen to satisfy (1.4). Then there exist constants
$\displaystyle\varepsilon_{0}\in(0,1)$, $\displaystyle\rho\in(0,1/2)$ and
$\displaystyle C_{0}>0$ depending on $\displaystyle\alpha$, $\displaystyle n$,
$\displaystyle\lambda$, $\displaystyle\Lambda$,
$\displaystyle\|D^{2}\phi\|_{L^{\infty}(\Omega)}$,
$\displaystyle\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}$, $\displaystyle
i(\Phi)$ and $\displaystyle s(\Phi)$ such that for any
$\displaystyle\xi\in\mathbb{R}^{n}$ and a viscosity solution, $\displaystyle
u$, of

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|Du|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
u(y)=g(y)&\text{on $\displaystyle
B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

the following holds: if

$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1\quad\text{and}\quad\|f\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0},$

then there exists an affine function $\displaystyle l(y)=a+b\cdot(y-x)$ with
$\displaystyle|a|+|b|\leq C_{0}$ such that for each $\displaystyle
0<r\leq\rho$,

$\displaystyle\displaystyle\|u-l\|_{L^{\infty}(B_{r}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
Cr^{1+\alpha}$

for some universal constant $\displaystyle C>0$.

Before we prove Lemma 5.1 by using the induction, we first show the
approximation lemma.

######  Lemma 5.2 (Approximation lemma; degenerate case).

Suppose (A1): -(A4):  hold true with $\displaystyle i(\Phi)\geq 0$ and
$\displaystyle\nu_{0}=\nu_{1}=1$. Let $\displaystyle\xi\in\mathbb{R}^{n}$ be
an arbitrary vector and $\displaystyle u\in
C(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})$ be a viscosity solution of

(5.3)
$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,|\xi+Du|)F(D^{2}u)=f(y)&\text{in
$\displaystyle B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
u(y)=g(y)&\text{on $\displaystyle
B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

satisfying
$\displaystyle\|u\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1$ and
$\displaystyle\|g\|_{C^{1,\beta_{g}}(B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\})}\leq
1$. Then for any $\displaystyle\mu>0$, there exists a constant
$\displaystyle\varepsilon_{0}=\varepsilon_{0}(n,\lambda,\Lambda,i(\Phi),L,\mu)>0$
such that if

$\displaystyle\displaystyle\|f\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0},$

then one can find a viscosity solution $\displaystyle h$ of an uniformly
$\displaystyle(\lambda,\Lambda)$-elliptic equation

(5.4) $\displaystyle\displaystyle\begin{cases}\mathcal{F}(D^{2}h)=0\ &\text{in
$\displaystyle B_{3/4}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\ h=u\ &\text{on
$\displaystyle B_{3/4}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$}\end{cases}$

such that

$\displaystyle\displaystyle\|u-h\|_{L^{\infty}(B_{1/2}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\mu.$

###### Proof.

By contradiction, we suppose the conclusion of the lemma fails. Therefore,
there exist $\displaystyle\mu_{0}>0$ and sequences of
$\displaystyle\\{F_{k}\\}_{k=1}^{\infty}$,
$\displaystyle\\{\Phi_{k}\\}_{k=1}^{\infty}$,
$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}$,
$\displaystyle\\{g_{k}\\}_{k=1}^{\infty}$, and
$\displaystyle\\{u_{k}\\}_{k=1}^{\infty}$ and a sequence of vectors
$\displaystyle\\{\xi_{k}\\}_{k=1}^{\infty}$ such that

(C1) :

$\displaystyle F_{k}\in C(\mathcal{S}(n),\mathbb{R})$ is uniformly
$\displaystyle(\lambda,\Lambda)$-elliptic;

(C2) :

for $\displaystyle\Phi_{k}\in C(B_{1}\times[0,\infty),[0,\infty))$, the map
$\displaystyle\textstyle t\mapsto\frac{\Phi_{k}(x,t)}{t^{i(\Phi)}}$ is almost
non-decreasing and the map $\displaystyle\textstyle
t\mapsto\frac{\Phi_{k}(x,t)}{t^{s(\Phi)}}$ is almost non-increasing with
constant $\displaystyle L\geq 1$, and $\displaystyle\Phi_{k}(y,1)=1$ for all
$\displaystyle y\in B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$;

(C3) :

$\displaystyle f_{k}\in C(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})$ with
$\displaystyle\|f_{k}\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1/k$;

(C4) :

$\displaystyle u_{k}\in C(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})$ with
$\displaystyle\|u\|_{L^{\infty}(B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1$ solves the equation

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi_{k}(y,|\xi_{k}+Du_{k}|)F_{k}(D^{2}u_{k})=f_{k}(y)&\text{in
$\displaystyle B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
u_{k}(y)=g_{k}(y)&\text{on $\displaystyle
B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

with
$\displaystyle\|g_{k}\|_{C^{1,\beta_{g}}(B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\})}\leq
1$, but

(5.5)
$\displaystyle\displaystyle\|u_{k}-h\|_{L^{\infty}(B_{1/2}(x)\cap\\{y_{n}>\phi(y^{\prime})\\})}>\mu_{0}\quad\text{for
any $\displaystyle k\in\mathbb{N}$},$

for any $\displaystyle h$ satisfying (5.4).

The condition (C1) :  implies that $\displaystyle F_{k}$ converges to some
uniformly $\displaystyle(\lambda,\Lambda)$-elliptic operator $\displaystyle
F_{\infty}\in C(\mathcal{S}(n),\mathbb{R})$. Similarly, the condition (C4) :
implies that $\displaystyle g_{k}$ converges to $\displaystyle g_{\infty}$
uniformly. For a further discussion, we consider two cases:

  1. (i)

($\displaystyle\\{\xi_{k}\\}_{k=1}^{\infty}$ is bounded) Up to a subsequence,
$\displaystyle\xi_{k}$ converges to some vector $\displaystyle\xi_{\infty}$.
Then we consider a sequence
$\displaystyle\\{\tilde{u}_{k}\\}_{k=1}^{\infty}:=\\{u_{k}+x\cdot\xi_{k}\\}_{k=1}^{\infty}$
satisfying

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi_{k}(y,|D\tilde{u}_{k}|)F(D^{2}\tilde{u}_{k})=f_{k}(y)&\text{in
$\displaystyle B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
\tilde{u}_{k}(y)=\tilde{g}_{k}(y)&\text{on $\displaystyle
B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$}\end{array}\right.$

for $\displaystyle\tilde{g}_{k}(x):=g_{k}(x)+x\cdot\xi_{k}$. Therefore, we can
apply Theorem 4.4 for $\displaystyle\tilde{u}_{k}$ and so by Arzela-Ascoli
theorem, we conclude that $\displaystyle u_{k}\to u_{\infty}$ uniformly in
$\displaystyle B_{r}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$ for any
$\displaystyle 0<r<1$. Indeed, $\displaystyle u_{\infty}$ satisfies

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi_{\infty}(y,|\xi_{\infty}+Du_{\infty}|)F(D^{2}u_{\infty})=0&\text{in
$\displaystyle B_{1}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
u_{\infty}(y)=g_{\infty}(y)&\text{on $\displaystyle
B_{1}(x)\cap\\{y_{n}=\phi(y^{\prime})\\}$}.\end{array}\right.$

Then by following the proof of [4, Lemma 4.1], we conclude that

$\displaystyle\displaystyle F_{\infty}(D^{2}u_{\infty})=0\quad\text{in
$\displaystyle B_{3/4}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},$

which leads to the contradiction with (5.5) (choose $\displaystyle
h=u_{\infty}$ and $\displaystyle\mathcal{F}=F_{\infty}$).

  2. (ii)

($\displaystyle\\{\xi_{k}\\}_{k=1}^{\infty}$ is unbounded) In this case, for
the constant $\displaystyle p_{0}>0$ chosen in Theorem 4.6, we may assume
$\displaystyle|\xi_{k}|>p_{0}$ and $\displaystyle|\xi_{k}|\to\infty$ (up to a
subsequence). Thus, we can apply Theorem 4.6 for $\displaystyle u_{k}$ and so
by Arzela-Ascoli theorem, we conclude that $\displaystyle u_{k}\to u_{\infty}$
uniformly in $\displaystyle B_{r}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$ for any
$\displaystyle 0<r<1$. Again by following the proof of [4, Lemma 4.1], we
conclude that

$\displaystyle\displaystyle F_{\infty}(D^{2}u_{\infty})=0\quad\text{in
$\displaystyle B_{3/4}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}$},$

which leads to the contradiction with (5.5) (choose $\displaystyle
h=u_{\infty}$ and $\displaystyle\mathcal{F}=F_{\infty}$).

∎

###### Proof of Lemma 5.1.

By the smallness regime in Remark 4.1, we may assume that $\displaystyle u\in
C(\Omega)$ is a viscosity solution with

$\displaystyle\displaystyle\|u\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
1,\,\|g\|_{C^{1,\beta_{g}}(B_{1}\cap\\{y_{n}=\phi(y^{\prime})\\})}\leq
1,\,\|f\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\varepsilon_{0},$

and $\displaystyle\nu_{0}=\nu_{1}=1$. As in [4, 8, 17], the proof is based on
the induction argument: we claim that there exist universal constants
$\displaystyle 0<\rho\ll 1$, $\displaystyle C_{0}>1$ and a sequence of affine
functions

$\displaystyle\displaystyle l_{k}(y):=a_{k}+b_{k}\cdot(y-x),$

where $\displaystyle\\{a_{k}\\}_{k=1}^{\infty}\subset\mathbb{R}$ and
$\displaystyle\\{b_{k}\\}_{k=1}^{\infty}\subset\mathbb{R}^{n}$ such that, for
every $\displaystyle k\in\mathbb{N}$,

(E1):

$\displaystyle\sup_{y\in
B_{\rho^{k}}(x)\cap\\{y_{n}>\phi(y^{\prime})\\}}|u(y)-l_{k}(y)|\leq\rho^{k(1+\alpha)}$;

(E2):

$\displaystyle|a_{k}-a_{k-1}|\leq C_{0}\rho^{(k-1)(1+\alpha)}$ and
$\displaystyle|b_{k}-b_{k-1}|\leq C_{0}\rho^{(k-1)\alpha}$.

  1. (i)

(Initial step) Without loss of generality, we may assume $\displaystyle x=0$.
Let us set

$\displaystyle\displaystyle l_{1}(y):=h(0)+Dh(0)\cdot y,$

where $\displaystyle h$ is the approximation function coming from Lemma 5.2
for a constant $\displaystyle\mu>0$ to be determined later. Then, by the
boundary estimate for uniformly elliptic fully nonlinear operators [3, 25],
there exists a constant $\displaystyle C_{0}=C_{0}(n,\lambda,\Lambda)>1$ such
that

$\displaystyle\displaystyle\|h\|_{C^{1,\alpha}(B_{1/2}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
C_{0}\quad\text{and}\quad\sup_{y\in
B_{\rho}\cap\\{y_{n}>\phi(y^{\prime})\\}}|h(y)-l_{1}(y)|\leq
C_{0}\rho^{1+\beta_{g}},$

for every $\displaystyle 0<\rho\leq 1/2$. The triangle inequality yields that

$\displaystyle\displaystyle\sup_{y\in
B_{\rho}\cap\\{y_{n}>\phi(y^{\prime})\\}}|u(y)-l_{1}(y)|\leq
C_{0}\rho^{1+\beta_{g}}+\mu.$

We now select a universal constant $\displaystyle 0<\rho\ll 1$ small enough so
that

$\displaystyle\displaystyle
C_{0}\rho^{1+\beta_{g}}\leq\frac{1}{2}\rho^{1+\alpha},\quad\rho^{\alpha}\leq\frac{1}{2},\quad\text{and}\quad\rho^{1-\alpha(1+s(\Phi))}\leq
1,$

which is possible due to the choice of $\displaystyle\alpha$; recall (1.4). In
a sequel, we choose a constant $\displaystyle\mu:=\rho^{1+\alpha}/2$ and set
$\displaystyle a_{0}=0,b_{0}=0$, $\displaystyle a_{1}=h(0)$ and $\displaystyle
b_{1}=Dh(0)$, which completes the proof of the initial step.

  2. (ii)

(Iterative procedure) We now suppose that (E1):  and (E2):  hold true for
$\displaystyle k\geq 1$. We then verify (E1):  and (E2):  for $\displaystyle
k+1$. For this purpose, we define a rescaled function

$\displaystyle\displaystyle
u_{k}(y):=\frac{u(\rho^{k}y)-l_{k}(\rho^{k}y)}{\rho^{k(1+\alpha)}}.$

Then $\displaystyle u_{k}$ satisfies

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\Phi_{k}(y,|\xi_{k}+Du_{k}|)F(D^{2}u_{k})=f_{k}(y)&\text{in
$\displaystyle B_{1}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\}$},\\\
u_{k}(y)=g_{k}(y)&\text{on $\displaystyle
B_{1}\cap\\{y_{n}=\phi_{k}(y^{\prime})\\}$},\end{array}\right.$

where

$\displaystyle\displaystyle F_{k}(M)$
$\displaystyle\displaystyle:=\rho^{k(1-\alpha)}F(\rho^{k(\alpha-1)}M),\quad\Phi_{k}(y,t):=\frac{\Phi(\rho^{k}y,\rho^{k\alpha}t)}{\Phi(\rho^{k}y,\rho^{k\alpha})},\quad
f_{k}(y):=\frac{\rho^{k(1-\alpha)}}{\Phi(\rho^{k}y,\rho^{k\alpha})}f(\rho^{k}y),$
$\displaystyle\displaystyle g_{k}(y)$
$\displaystyle\displaystyle:=\frac{g(\rho^{k}y)-l_{k}(\rho^{k}y)}{\rho^{k(1+\alpha)}},\quad\phi_{k}(y^{\prime}):=\rho^{-k}\phi(\rho^{k}y^{\prime})\text{
and }\xi_{k}:=\rho^{-k\alpha}b_{k}.$

It can be easily checked that

    1. (a)

$\displaystyle F_{k}$ satisfies (A1):  with the same constants
$\displaystyle(\lambda,\Lambda)$;

    2. (b)

$\displaystyle\Phi_{k}$ satisfies (A2):  with the same constants
$\displaystyle(i(\Phi),s(\Phi))$ and $\displaystyle\Phi_{k}(y,1)\equiv 1$;

    3. (c)

$\displaystyle\|u_{k}\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\})}\leq
1$ by the induction hypothesis;

    4. (d)

$\displaystyle\|f_{k}\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\})}\leq
L\varepsilon_{0}\rho^{k(1-\alpha(1+s(\Phi)))}\leq L\varepsilon_{0}$;

    5. (e)

$\displaystyle\|D^{2}\phi_{k}\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\})}\leq\rho^{k}\|D^{2}\phi\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq\|D^{2}\phi\|_{L^{\infty}(B_{1}\cap\\{y_{n}>\phi(y^{\prime})\\})}$.

Moreover, for $\displaystyle g_{k}$, we can compute

$\displaystyle\displaystyle|Dg_{k}(y)-Dg_{k}(z)|=\rho^{-k\alpha}|Dg(\rho^{k}y)-Dg(\rho^{k}z)|$
$\displaystyle\displaystyle\leq\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}\cdot\rho^{-k\alpha}|\rho^{k}(y-z)|^{\beta_{g}}$
$\displaystyle\displaystyle\leq\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}\cdot|y-z|^{\beta_{g}}.$

By recalling the fact that $\displaystyle u=g$ on
$\displaystyle\partial\Omega$ together with the induction hypothesis, we can
conclude that

$\displaystyle\displaystyle\|g_{k}\|_{C^{1,\beta_{g}}(B_{1}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\})}\leq\|g\|_{C^{1,\beta_{g}}(\partial\Omega)}\leq
1.$

Hence, we now apply Lemma 5.2 for $\displaystyle u_{k}$ and then follow the
argument in the initial step to ensure the existence of an affince function
$\displaystyle\overline{l}(y):=\overline{a}+\overline{b}\cdot y$ such that

$\displaystyle\displaystyle\sup_{y\in
B_{\rho}\cap\\{y_{n}>\phi_{k}(y^{\prime})\\}}|u_{k}(y)-\overline{l}(y)|\leq\rho^{1+\alpha}\quad\text{and}\quad|\overline{a}|,|\overline{b}|\leq
C_{0}.$

By scaling back, we conclude that

$\displaystyle\displaystyle\sup_{y\in
B_{\rho^{k+1}}\cap\\{y_{n}>\phi(y^{\prime})\\}}|u(y)-l_{k+1}(y)|\leq\rho^{(k+1)(1+\alpha)},$

where

$\displaystyle\displaystyle
l_{k+1}(y):=l_{k}(y)+\rho^{k(1+\alpha)}\cdot\overline{l}(\rho^{-k}y).$

Here note that

$\displaystyle\displaystyle|a_{k+1}-a_{k}|$
$\displaystyle\displaystyle=\rho^{k(1+\alpha)}|\overline{a}|\leq
C_{0}\rho^{k(1+\alpha)},$ $\displaystyle\displaystyle|b_{k+1}-b_{k}|$
$\displaystyle\displaystyle=\rho^{k\alpha}|\overline{b}|\leq
C_{0}\rho^{k\alpha}.$

Therefore, (E1):  and (E2):  hold for $\displaystyle k+1$.

∎

###### Proof of Theorem 1.1.

We first consider the degenerate case, i.e., $\displaystyle i(\Phi)\geq 0$. By
Lemma 5.1, a viscosity solution can be approximated by an affine function with
an error of order $\displaystyle r^{1+\alpha}$. By following the argument in
[8], we can derive the desired global $\displaystyle C^{1,\alpha}$-estimate
provided that $\displaystyle i(\Phi)\geq 0$.

On the other hand, for the singular case ($\displaystyle i(\Phi)<0$), we
employ the idea of [4]. Indeed, we claim that Lemma 5.1 still holds for the
singular case. Theorem 4.4 guarantees that

$\displaystyle\displaystyle\|u\|_{C^{0,1}(B_{3/4}\cap\\{y_{n}>\phi(y^{\prime})\\})}\leq
c,$

for a universal constant $\displaystyle c>0$. Then $\displaystyle u$ is a
viscosity solution of

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\tilde{\Phi}(y,|Du|)F(D^{2}u)=\tilde{f}(y)&\text{in
$\displaystyle B_{3/4}\cap\\{y_{n}>\phi(y^{\prime})\\}$},\\\
u(y)=g(y)&\text{on $\displaystyle
B_{3/4}\cap\\{y_{n}=\phi(y^{\prime})\\}$},\end{array}\right.$

where

$\displaystyle\displaystyle\tilde{\Phi}(y,t):=t^{-i(\Phi)}\Phi(y,t)\quad\text{and}\quad\tilde{f}(y):=|Du|^{-i(\Phi)}f(y).$

Here $\displaystyle\tilde{\Phi}$ satisfies the condition (A2):  with
$\displaystyle i(\tilde{\Phi})=0$, $\displaystyle
s(\tilde{\Phi})=s(\Phi)-i(\Phi)$, and

$\displaystyle\displaystyle\|\tilde{f}\|_{L^{\infty}(\Omega)}\leq
c^{-i(\Phi)}\varepsilon_{0}.$

Thus, one can repeat the argument in the proof of Lemma 5.1 to obtain the
global $\displaystyle C^{1,\alpha}$-estimate. ∎

##  6\. Comparison principle and Perron’s method

The purpose of this section is to study the classical result of comparison
principle and as a consequence, to deduce the existence of a viscosity
solution to (1.3) by Perron’s method. Nevertheless, the assumptions (A1):
-(A4):  are not sufficient to obtain the aforementioned results. Therefore, we
require an additional assumption (A5):  which guarantees the comparison
principle for approximated Dirichlet problems. Before we precisely state this
new assumption, we summarize known results regarding the comparison principle.

######  Remark 6.1 (Comparison principle).

Let $\displaystyle
H:\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathcal{S}^{n}\to\mathbb{R}$
be a proper map. In other words, $\displaystyle H$ satisfies

(6.1) $\displaystyle\displaystyle H(x,r,p,X)$ $\displaystyle\displaystyle\leq
H(x,r,p,Y)\quad\text{whenever $\displaystyle X\leq Y$},$ (6.2)
$\displaystyle\displaystyle H(x,r,p,X)$ $\displaystyle\displaystyle\leq
H(x,s,p,X)\quad\text{whenever $\displaystyle s\leq r$.}$

Then we say that $\displaystyle H$ satisfies the comparison principle if the
following holds:

Let $\displaystyle v\in\mathrm{USC}(\overline{\Omega})$ [resp. $\displaystyle
w\in\mathrm{LSC}(\overline{\Omega})$] be a subsolution [resp. supersolution]
of $\displaystyle H=0$ in $\displaystyle\Omega$ and $\displaystyle v\leq w$ on
$\displaystyle\partial\Omega$. Then $\displaystyle v\leq w$ in
$\displaystyle\overline{\Omega}$.

We refer to [5, 10, 19, 23, 24] for several sufficient conditions of then
comparison principle. In short, $\displaystyle H$ satisfies the comparison
principle if $\displaystyle H(x,r,p,X)$ is independent of $\displaystyle x$,
and one of the following conditions holds:

  1. (i)

$\displaystyle H(x,r,p,X)$ is strictly decreasing in $\displaystyle r$ and
$\displaystyle H$ is degenerate elliptic (i.e., $\displaystyle H$ satisfies
(6.1)), or

  2. (ii)

$\displaystyle H(x,r,p,X)$ is nonincreasing in $\displaystyle r$ and
$\displaystyle H$ is uniformly elliptic.

It is noteworthy that the condition that $\displaystyle H$ is independent of
$\displaystyle x$ can be relaxed to some extra structure conditions on
$\displaystyle H$, which display a kind of smoothness on $\displaystyle H$
with respect to $\displaystyle x$-variable; see [10, 19, 23] for details.

We now consider a proper map

$\displaystyle\displaystyle H(x,u,Du,D^{2}u):=\Phi(x,|Du|)F(D^{2}u)-f(x)$

with (A1): -(A4): . It is easily checked that $\displaystyle H$ is degenerate
elliptic, nonincreasing in $\displaystyle r$, and $\displaystyle H$ depends on
$\displaystyle x$. In view of the previous remark, we cannot expect the
comparison principle for $\displaystyle H$ and Perron’s method for the
associated Dirichlet problem. To overcome this challenge, we will impose an
additional structure condition on $\displaystyle H$ and approximate the map
$\displaystyle H$ to ensure the strict monotonicity with respect to
$\displaystyle r$-variable:

  1. (i)

(A5):

There exists a continuous function
$\displaystyle\omega:[0,\infty)\to[0,\infty)$ such that
$\displaystyle\omega(0)=0$ and

$\displaystyle\displaystyle\Phi(x,\alpha|x-y|)F(X)-\Phi(y,\alpha|x-y|)F(-Y)\leq\omega(\alpha|x-y|^{2}+|x-y|),$

whenever $\displaystyle\alpha>0$, $\displaystyle x,y\in\Omega$, $\displaystyle
X,Y\in\mathcal{S}^{n}$ and

(6.3) $\displaystyle\displaystyle-3\alpha\begin{pmatrix}I&0\\\
0&I\end{pmatrix}\leq\begin{pmatrix}X&0\\\ 0&Y\end{pmatrix}\leq
3\alpha\begin{pmatrix}I&-I\\\ -I&I\end{pmatrix}.$

Similar conditions with (A5):  can be found in [5, 10].

  2. (ii)

Let us consider the approximated problem given by

(6.4) $\displaystyle\displaystyle
H_{\varepsilon}(x,u,Du,D^{2}u):=\Phi(x,|Du|)F(D^{2}u)-f(x)-\varepsilon
u=0\quad\text{in}\quad\Omega$

for $\displaystyle\varepsilon>0$. Clearly, $\displaystyle H_{\varepsilon}$ is
strictly decreasing in $\displaystyle r$.

######  Lemma 6.2 (Comparison principle II).

Suppose that the assumptions (A1): , (A2): , (A5):  are in force and
$\displaystyle f\in C(\overline{\Omega})$. Then $\displaystyle
H_{\varepsilon}$ satisfies the comparison principle:

Let $\displaystyle v$ and $\displaystyle w$ be a viscosity subsolution and a
supersolution of (6.4), respectively. If $\displaystyle v\leq w$ on
$\displaystyle\partial\Omega$, then $\displaystyle v\leq w$ in
$\displaystyle\Omega$.

###### Proof.

By contradiction, we suppose that

$\displaystyle\displaystyle
L_{0}:=\sup\limits_{x\in\overline{\Omega}}(v(x)-w(x))>0.$

For any $\displaystyle\alpha>0$, we define

$\displaystyle\displaystyle
L_{\alpha}:=\sup\limits_{x,y\in\overline{\Omega}}\left[v(x)-w(y)-(\alpha/2)|x-y|^{2}\right]$

and clearly $\displaystyle L_{\alpha}\geq L_{0}$. Suppose that the maximum
$\displaystyle L_{\alpha}$ is attained at a point
$\displaystyle(x_{\alpha},y_{\alpha})\in\overline{\Omega}\times\overline{\Omega}$.
It implies from [10, Lemma 3.1] that

$\displaystyle\displaystyle\lim\limits_{\alpha\to\infty}\alpha|x_{\alpha}-y_{\alpha}|^{2}=0.$

This one and the fact that $\displaystyle v\leq w$ on
$\displaystyle\partial\Omega$ yield that $\displaystyle
x_{\alpha},y_{\alpha}\in\Omega$ for $\displaystyle\alpha>0$ large enough. At
this moment, we are able to apply Ishii-Lions lemma, [10, Theorem 3.2], to
ensure that there exist a limiting super-jet
$\displaystyle\left(\alpha(x_{\alpha}-y_{\alpha}),X_{\alpha}\right)$ of
$\displaystyle v$ at $\displaystyle x_{\alpha}$ and a limiting sub-jet
$\displaystyle\left(\alpha(x_{\alpha}-y_{\alpha}),-Y_{\alpha}\right)$ of
$\displaystyle w$ at $\displaystyle y_{\alpha}$ so that

$\displaystyle\displaystyle-3\alpha\begin{pmatrix}I&0\\\
0&I\end{pmatrix}\leq\begin{pmatrix}X_{\alpha}&0\\\
0&Y_{\alpha}\end{pmatrix}\leq 3\alpha\begin{pmatrix}I&-I\\\ -I&I\end{pmatrix}$

and

$\displaystyle\displaystyle\begin{cases}\Phi\left(x_{\alpha},\alpha|x_{\alpha}-y_{\alpha}|\right)F(X_{\alpha})-\varepsilon
v(x_{\alpha})\geq f(x_{\alpha}),\\\
\Phi\left(y_{\alpha},\alpha|x_{\alpha}-y_{\alpha}|\right)F(-Y_{\alpha})-\varepsilon
w(y_{\alpha})\leq f(y_{\alpha}).\end{cases}$

By using the relation $\displaystyle
L_{0}=\lim_{\alpha\to\infty}[v(x_{\alpha})-w(y_{\alpha})]$ and the assumption
(A5): , we have, for sufficiently large $\displaystyle\alpha>0$,

$\displaystyle\displaystyle\frac{\varepsilon L_{0}}{2}$
$\displaystyle\displaystyle\leq\varepsilon[v(x_{\alpha})-w(y_{\alpha})]$
$\displaystyle\displaystyle\leq
f(y_{\alpha})-f(x_{\alpha})+\Phi\left(x_{\alpha},\alpha|x_{\alpha}-y_{\alpha}|\right)F(X_{\alpha})-\Phi\left(y_{\alpha},\alpha|x_{\alpha}-y_{\alpha}|\right)F(-Y_{\alpha})$
$\displaystyle\displaystyle\leq
f(y_{\alpha})-f(x_{\alpha})+\omega(\alpha|x_{\alpha}-y_{\alpha}|^{2}+|x_{\alpha}-y_{\alpha}|).$

Since $\displaystyle f\in C(\overline{\Omega})$ and
$\displaystyle\omega(0+)=0$, we arrive at a contradiction when
$\displaystyle\alpha\to\infty$. ∎

We now turn our attention to showing the existence of a viscosity solution to
(6.4).

######  Lemma 6.3 (Existence of sub/supersolutions).

Suppose the assumptions (A1): -(A4):  are in force. Then for every
$\displaystyle\varepsilon\in(0,1)$, there exist a viscosity subsolution
$\displaystyle v_{\varepsilon}\in C(\overline{\Omega})$ and a viscosity
supersolution $\displaystyle w_{\varepsilon}\in C(\overline{\Omega})$ of (6.4)
with $\displaystyle v_{\varepsilon}=w_{\varepsilon}=g$ on
$\displaystyle\partial\Omega$. Moreover, there exists a positive constant
$\displaystyle c\equiv
c(n,\lambda,\Lambda,\nu_{0},L,r,\mathrm{diam}(\Omega),\|f\|_{L^{\infty}(\Omega)},\|g\|_{L^{\infty}(\partial\Omega)})$
such that

$\displaystyle\displaystyle-c\leq v_{\varepsilon}\leq w_{\varepsilon}\leq
c,\quad\text{for any $\displaystyle 0<\varepsilon<1$.}$

###### Proof.

Let $\displaystyle z\in\partial\Omega$ be a fixed point. There exists a point
$\displaystyle x_{z}\in\mathbb{R}^{n}\setminus\overline{\Omega}$ such that
$\displaystyle\overline{B_{r}(x_{z})}\cap\overline{\Omega}=\\{z\\}$ with
$\displaystyle r=|z-x_{z}|$ since $\displaystyle\Omega$ satisfies a uniform
exterior sphere property; see Lemma 2.2. We now consider a function
$\displaystyle v_{z}:\overline{\Omega}\rightarrow[0,\infty)$ defined by

$\displaystyle\displaystyle
v_{z}(x):=K\left(r^{-\kappa_{0}}-|x-x_{z}|^{-\kappa_{0}}\right)$

for positive constants $\displaystyle\kappa_{0}:=\frac{n\Lambda+1}{\lambda}$
and $\displaystyle
K\geq\min\left\\{1,\frac{R^{\kappa_{0}+1}}{\kappa_{0}}\right\\}$ to be
determined later, with $\displaystyle R:=r+\mathrm{diam}(\Omega)$. Note that
$\displaystyle v_{z}(z)=0$, $\displaystyle v_{z}>0$ in $\displaystyle\Omega$,
and direct calculations yield that

$\displaystyle\displaystyle
Dv_{z}(x)=K\kappa_{0}\frac{x-x_{z}}{|x-x_{z}|^{\kappa_{0}+2}}$

and

$\displaystyle\displaystyle
D^{2}v_{z}(x)=K\kappa_{0}\frac{I}{|x-x_{z}|^{\kappa_{0}+2}}-K\kappa_{0}(\kappa_{0}+2)\frac{(x-x_{z})\otimes(x-x_{z})}{|x-x_{z}|^{\kappa_{0}+4}}.$

Due to the choice of $\displaystyle\kappa_{0}$, we have

$\displaystyle\displaystyle
F(D^{2}v_{z}(x))\leq\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+2}}\left((n-1)\Lambda-(\kappa_{0}+1)\lambda\right)\leq-\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+2}}.$

On the other hand, for a fixed $\displaystyle\delta\in(0,1)$, we further
define

$\displaystyle\displaystyle v_{z,\delta}(x):=g(z)+\delta+M_{\delta}v_{z}(x),$

where the constant $\displaystyle M_{\delta}\geq 1$ can be chosen so that
$\displaystyle v_{z,\delta}\geq g$ on $\displaystyle\partial\Omega$. This is
possible because $\displaystyle K\geq 1$ and $\displaystyle g$ is continuous
on $\displaystyle\partial\Omega$. Indeed, $\displaystyle M_{\delta}$ depends
only on the modulus of continuity of $\displaystyle g$, and is independent of
$\displaystyle z$. Then we see that

$\displaystyle\displaystyle\Phi(x,|Dv_{z,\delta}(x)|)F(D^{2}v_{z,\delta}(x))-\varepsilon
v_{z,\delta}(x)$
$\displaystyle\displaystyle\leq\frac{\nu_{0}}{L}\min\left\\{\left(\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+1}}\right)^{i(\Phi)},\left(\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+1}}\right)^{s(\Phi)}\right\\}\cdot\left(-M_{\delta}\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+2}}\right)+\varepsilon\left\lVert
g\right\rVert_{L^{\infty}(\partial\Omega)}.$

Here if we let $\displaystyle
A:=\frac{K\kappa_{0}}{|x-x_{z}|^{\kappa_{0}+1}}$, then

$\displaystyle\displaystyle\min\left\\{A^{i(\Phi)},A^{s(\Phi)}\right\\}$
$\displaystyle\displaystyle=A^{-1}\min\left\\{A^{i(\Phi)+1},A^{s(\Phi)+1}\right\\}$
$\displaystyle\displaystyle\geq\left(\frac{K\kappa_{0}}{r^{\kappa_{0}+1}}\right)^{-1}\min\left\\{\left(\frac{K\kappa_{0}}{R^{\kappa_{0}+1}}\right)^{i(\Phi)+1},\left(\frac{K\kappa_{0}}{R^{\kappa_{0}+1}}\right)^{s(\Phi)+1}\right\\}.$

Note that we need to select $\displaystyle K$ so that
$\displaystyle\frac{K\kappa_{0}}{R^{\kappa_{0}+1}}\geq 1$. Since
$\displaystyle 0<i(\Phi)+1\leq s(\Phi)+1$, we conclude that

$\displaystyle\displaystyle\min\left\\{A^{i(\Phi)},A^{s(\Phi)}\right\\}\geq\left(\frac{\kappa_{0}}{r^{\kappa_{0}+1}}\right)^{-1}\left(\frac{\kappa_{0}}{R^{\kappa_{0}+1}}\right)^{i(\Phi)+1}K^{i(\Phi)}.$

Hence, we deduce that

$\displaystyle\displaystyle\Phi(x,|Dv_{z,\delta}(x)|)$
$\displaystyle\displaystyle F(D^{2}v_{z,\delta}(x))-\varepsilon
v_{z,\delta}(x)$
$\displaystyle\displaystyle\leq-\frac{\nu_{0}}{L}\left(\frac{\kappa_{0}}{r^{\kappa_{0}+1}}\right)^{-1}\left(\frac{\kappa_{0}}{R^{\kappa_{0}+1}}\right)^{i(\Phi)+1}\frac{\kappa_{0}}{R^{\kappa_{0}+2}}K^{i(\Phi)+1}+\left\lVert
g\right\rVert_{L^{\infty}(\partial\Omega)}.$

Therefore, we can choose $\displaystyle
K=K(n,\lambda,\Lambda,\nu_{0},L,r,\mathrm{diam}(\Omega),\|f\|_{L^{\infty}(\Omega)},\|g\|_{L^{\infty}(\partial\Omega)})$
large enough so that

$\displaystyle\displaystyle\Phi(x,|Dv_{z,\delta}(x)|)$
$\displaystyle\displaystyle F(D^{2}v_{z,\delta}(x))-\varepsilon
v_{z,\delta}(x)\leq-\|f\|_{L^{\infty}(\Omega)}\quad\text{in
$\displaystyle\Omega$},$

i.e., $\displaystyle v_{z,\delta}$ is a viscosity supersolution to (6.4).

Finally, we define

$\displaystyle\displaystyle
w_{\varepsilon}(x):=\inf\left\\{v_{z,\delta}(x)\text{: $\displaystyle
z\in\partial\Omega$ and $\displaystyle\delta\in(0,1)$}\right\\}.$

It is easy to check that $\displaystyle w_{\varepsilon}$ is a viscosity
supersolution to (6.4) in $\displaystyle\Omega$ and enjoys the boundary
condition $\displaystyle w_{\varepsilon}=g$ on $\displaystyle\partial\Omega$.
Moreover, it immediately follows from the construction of $\displaystyle
w_{\varepsilon}$ that

$\displaystyle\displaystyle w_{\varepsilon}\leq
C(n,\lambda,\Lambda,\nu_{0},L,\mathrm{diam}(\Omega),\|f\|_{L^{\infty}(\Omega)},\|g\|_{L^{\infty}(\partial\Omega)})\quad\text{in
$\displaystyle\Omega$},$

for any $\displaystyle\varepsilon\in(0,1)$. The existence of a viscosity
subsolution $\displaystyle v_{\varepsilon}$ and its lower bound can be shown
in a similar argument. Finally, since $\displaystyle
v_{\varepsilon}=w_{\varepsilon}=g$ on $\displaystyle\partial\Omega$, Lemma 6.2
implies that $\displaystyle v_{\varepsilon}\leq w_{\varepsilon}$ in
$\displaystyle\Omega$. ∎

###### Proof of Theorem 1.2.

An application of Perron’s method [10, Theorem 4.1] together with Lemma 6.2
and Lemma 6.3 yields the existence of a viscosity solution $\displaystyle
u_{\varepsilon}$ to the approximated equation (6.4) with the boundary
condition $\displaystyle u_{\varepsilon}=g$.

We now understand $\displaystyle u_{\varepsilon}$ as a viscosity solution of

$\displaystyle\displaystyle\left\\{\begin{array}[]{rclcc}\Phi(x,|Du_{\varepsilon}|)F(D^{2}u_{\varepsilon})&=&f_{\varepsilon}(x)&\mbox{in
}&\Omega,\\\ u_{\varepsilon}(x)&=&g(x)&\mbox{on
}&\partial\Omega,\end{array}\right.$

where $\displaystyle f_{\varepsilon}(x):=f(x)+\varepsilon u_{\varepsilon}(x)$.
Here note that $\displaystyle\\{f_{\varepsilon}\\}_{\varepsilon\in(0,1)}$ is
uniformly bounded in $\displaystyle L^{\infty}(\Omega)$ by Lemma 6.3. Then by
applying [4, Lemma 3.1] and Theorem 4.4 (or just by applying the stronger
result Theorem 1.1), we have that
$\displaystyle\\{u_{\varepsilon}\\}_{\varepsilon\in(0,1)}$ is uniformly
bounded in $\displaystyle C^{0,\gamma}(\overline{\Omega})$ for some
$\displaystyle\gamma\in(0,1)$. Therefore, we can extract a uniformly
converging subsequence such that $\displaystyle u_{\varepsilon_{j}}\to
u_{\infty}$ when $\displaystyle\varepsilon_{j}\to 0$, and by the standard
stability argument, we conclude that $\displaystyle u_{\infty}$ solves (1.3).
∎

Conflict of interest. The authors declare that they have no conflict of
interest.

Acknowledgments. This work is supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MSIP): NRF-2021R1A4A1027378.
Se-Chan Lee is supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (2022R1A6A3A01086546).

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