††S.-S. Byun was supported by NRF-2017R1A2B2003877. J. Han was supported by
NRF-2019R1C1C1003844. J.Oh was supported by NRF-2020R1C1C1A01014904.

# On $W^{2,p}$-estimates for solutions of obstacle problems for fully
nonlinear elliptic equations with oblique boundary conditions

Sun-Sig Byun  Department of Mathematical Sciences, Seoul National University,
Seoul 08826, Republic of Korea<EMAIL_ADDRESS>,  Jeongmin Han  Department of
Mathematical Sciences, Seoul National University, Seoul 08826, Republic of
Korea<EMAIL_ADDRESS>and  Jehan Oh  Department of Mathematics, Kyungpook
National University, Daegu 41566, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

This paper concerns fully nonlinear elliptic obstacle problems with oblique
boundary conditions. We investigate the existence, uniqueness and
$W^{2,p}$-regularity results by finding approximate non-obstacle problems with
the same oblique boundary condition and then making a suitable limiting
process.

###### Key words and phrases:

$W^{2,p}$-regularity; Fully nonlinear equations; Oblique derivative problems;
Obstacle problems

###### 2010 Mathematics Subject Classification:

Primary: 35J25; Secondary: 35J15, 35J60.

##  1\. Introduction and main results

This paper concerns the existence, uniqueness and regularity for viscosity
solutions to the following obstacle problem with oblique boundary data

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u,Du,u,x)\leq f&\textrm{in
$\Omega,$}\\\ (F(D^{2}u,Du,u,x)-f)(u-\psi)=0&\textrm{in $\Omega$, }\\\
u\geq\psi&\textrm{in $\Omega$,}\\\ \beta\cdot Du=0&\textrm{on
$\partial\Omega$}\end{array}\right.$ (1.5)

for a given obstacle $\psi\in W^{2,p}(\Omega)$ satisfying $\beta\cdot
D\psi\geq 0$ a.e. on $\partial\Omega$. Here $\Omega$ is a bounded domain in
$\mathbb{R}^{n}$ with its boundary $\partial\Omega\in C^{3}$, $F$ is uniformly
elliptic with constants $\lambda$ and $\Lambda$, i.e.,

$\lambda||N||\leq F(M+N,q,r,x)-F(M,q,r,x)\leq\Lambda||N||$

for any $n\times n$ symmetric matrices $M,N$ with $N\geq 0$,
$q\in\mathbb{R}^{n}$, $r\in\mathbb{R}$ and $x\in\Omega$, and $\beta$ is a
vector-valued function with $||\beta||_{L^{\infty}(\partial\Omega)}=1$ and
$\beta\cdot\mathbf{n}\geq\delta_{0}$ for some positive constant $\delta_{0}$,
where $\mathbf{n}$ is the inner unit normal vector field of $\partial\Omega$.

The main purpose of this paper is to find an optimal $W^{2,p}$-regularity
theory for (1.5). More precisely, we want to identify the minimal condition of
$F$ with respect to $x$-variable under which the Hessian of a solution is as
integrable as both the nonhomogeneous term $f$ and the Hessian of the obstacle
$\psi$ in the setting of $L^{p}$ spaces for $n<p<\infty$.

Throughout this paper, we assume that $F=F(X,q,r,x)$ is convex in $X$ and
satisfies

$\displaystyle d(r_{2}-r_{1})\leq F(X,q,r_{1},x)-F(X,q,r_{2},x)$ (1.6)

for any $X\in S(n)$, $q\in\mathbb{R}^{n}$, $r_{1},r_{2}\in\mathbb{R}$ with
$r_{1}\leq r_{2}$, $x\in\Omega$, and some $d>0$. We further assume that

$\displaystyle\begin{split}&\mathcal{M}^{-}(\lambda,\Lambda,X_{1}-X_{2})-b|q_{1}-q_{2}|-c|r_{1}-r_{2}|\\\
&\qquad\leq F(X_{1},q_{1},r_{1},x)-F(X_{2},q_{2},r_{2},x)\\\
&\qquad\qquad\leq\mathcal{M}^{+}(\lambda,\Lambda,X_{1}-X_{2})+b|q_{1}-q_{2}|+c|r_{1}-r_{2}|\end{split}$
(1.7)

for $X_{1},X_{2}\in S(n)$, $q_{1},q_{2}\in\mathbb{R}^{n}$,
$r_{1},r_{2}\in\mathbb{R}$ and $x\in\Omega$. These assumptions are essential
in order to derive our desired results for solutions of (1.5), such as the
existence, uniqueness and $W^{2,p}$-regularity.

With an oscillation function of $F$ defined as

$\displaystyle\Theta_{F}(x_{1},x_{2}):=\sup_{X\in
S(n)\backslash\\{0\\}}\frac{|F(X,0,0,x_{1})-F(X,0,0,x_{2})|}{||X||}$

alongside a small perturbation of $\Theta_{F}$ from its integral average in
the $L^{n}$-sense, we shall prove the $W^{2,p}$-regularity for (1.5), as we
now state the main result of the paper. We remark that this approach to derive
$W^{2,p}$-regularity was employed in [Caf89].

######  Theorem 1.1.

Let $n<p<\infty$. Assume that $F=F(X,q,r,x)$ is convex in $X$, satisfies
(1.6)-(1.7) and $F(0,0,0,x)\equiv 0$, $\partial\Omega\in C^{3},f\in
L^{p}(\Omega),\beta\in C^{2}(\partial\Omega)$ with
$\beta\cdot\mathbf{n}\geq\delta_{0}$ for some $\delta_{0}>0$ and $\psi\in
W^{2,p}(\Omega)$. Then there exists a small
$\epsilon=\epsilon(n,\lambda,\Lambda,p,\delta_{0},||\beta||_{C^{2}(\partial\Omega)})>0$
such that if

$\displaystyle\sup_{x_{0}\in\overline{\Omega},0<\rho<\rho_{0}}\bigg{(}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}\Theta_{F}(x,x_{0})^{n}\
dx\bigg{)}^{1/n}\leq\epsilon$ (1.8)

for some $\rho_{0}>0$, then there is a unique viscosity solution $u\in
W^{2,p}(\Omega)$ of (1.5) with the following estimate

$||u||_{W^{2,p}(\Omega)}\leq
c(||f||_{L^{p}(\Omega)}+||\psi||_{W^{2,p}(\Omega)})$

for some constant
$c=c(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\partial\Omega,\operatorname{diam}(\Omega),\rho_{0})$.

One of the important issues regarding the obstacle problem is to study
solutions near the boundary of the contact surface with the obstacle. To this
end, suitable approximation methods have been used. In this regard, we revisit
the argument made in [BLOP18] where the Dirichlet boundary problem was studied
instead. Our main difficulty in using such an argument comes from the
situation that we are treating here the oblique boundary condition. Thus, we
need to modify the tools used in [BLOP18] properly to derive the desired
boundary estimates in the present paper. To do this, we verify several uniform
properties of a solution for the corresponding non-obstacle problem such as
$W^{2,p}$-regularity and comparison principle.

As a generalization of Neumann boundary problems, researches on oblique
derivative problems have been extensively made as in [Lie84, LT86, Lie87,
Saf95, DFP96, PRS00]. In particular, several notable results for fully
nonlinear elliptic equations were obtained in the notion of viscosity
solutions. The existence and uniqueness of fully nonlinear oblique derivative
problems were proved in [DI90, DI91, Ish91]. For the regularity of the
associated limiting problem, there have been established
$C^{2,\alpha}$-estimates, see [MS06] for the Neumann boundary condition and
[LZ18] for the oblique boundary condition, respectively. In [BH20] a global
$W^{2,p}$-regularity for the elliptic oblique derivative problem was proved.

On the other hand, the obstacle problem has been studied along with the free
boundary problem. We refer the reader to [Fri82, PSU12] for a general theory
of the obstacle problem. Regularity results for the elliptic obstacle problem
can be found in [FS14, ALS15, IM16, BLOP19]. We would like to point out that
$C^{1}$-regularity for the obstacle problem with the oblique boundary
condition was shown in [Lie01], while $W^{2,p}$-regularity for the Dirichlet
obstacle problem was given in [BLOP18]. The main purpose of this paper is to
derive a $W^{2,p}$-estimate for the oblique derivative problem with a
$W^{2,p}$-obstacle.

The remaining part of the paper is organized in the following way. In the next
section we introduce basic notation and give a brief exposition of viscosity
solutions. Section 3 deals with the associated oblique derivative problem
without obstacles. In particular we discuss the existence, uniqueness and
$W^{2,p}$-regularity for the non-obstacle problem. In the last section we
finally give the proof of Theorem 1.1.

##  2\. Preliminaries

###  2.1. Notations

We first introduce some notations which will be used in this paper.

  * •

$B_{r}(x_{0}):=\\{x\in\mathbb{R}^{n}:|x-x_{0}|<r\\}$ for
$x_{0}\in\mathbb{R}^{n}$, $r>0$. $B_{r}=B_{r}(0)$.

  * •

$S(n)$ is the set of $n\times n$ symmetric matrices and $||M||=\sup_{|x|\leq
1}|Mx|$ for any $M\in S(n)$.

  * •

We denote the gradient and Hessian of $u$ by $Du=(D_{1}u,\cdots,D_{n}u)$ and
$D^{2}u=(D_{ij}u)$, respectively. Here $D_{i}u=\frac{\partial u}{\partial
x_{i}}$ and $D_{ij}u=\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}$ for
$1\leq i,j\leq n$.

  * •

For any measurable set $A$ with $|A|\neq 0$ and measurable function $f$, to
mean the integral average of $f$ over $A$,

$\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptA}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}f\
dx=\frac{1}{|A|}\int_{A}f\ dx.$

###  2.2. Basic concepts

In this subsection, we briefly present some background knowledge for our
discussion. As usual, we are treating a viscosity solution. To do this, we
consider the following problem with oblique boundary data

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u,Du,u,x)=f&\textrm{in
$\Omega,$}\\\ \beta\cdot Du=0&\textrm{on $\partial\Omega$,}\\\
\end{array}\right.$ (2.3)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain. There are several
ways to define a viscosity solution depending on the choice of a test
function. In this paper, we take a test function $\varphi$ in
$W^{2,p}(\Omega)$. The solution defined in this way is called an
$L^{p}$-viscosity solution.

######  Definition 2.1.

Let $F$ be continuous in $X$ and measurable in $x$. Suppose $q>n$ and $f\in
L^{q}(\Omega)$. A continuous function $u$ is called an $L^{q}$-viscosity
solution for (2.3) if the following conditions hold:

  * (a)

(subsolution) For each $\varphi\in W^{2,q}(\Omega)$, whenever $\epsilon>0$,
$\mathcal{O}$ is relatively open in $\overline{\Omega}$ and

$F(D^{2}\varphi(x),D\varphi(x),\varphi(x),x)\leq
f(x)-\epsilon\quad\textrm{a.e. in}\ \mathcal{O}$

and

$\beta\cdot D\varphi(x)\leq-\epsilon\quad\textrm{a.e. on}\
\mathcal{O}\cap\partial\Omega,$

$u-\varphi$ cannot attain a local maximum in $\mathcal{O}$.

  * (b)

(supersolution) For each $\varphi\in W^{2,q}(\Omega)$, whenever $\epsilon>0$,
$\mathcal{O}$ is relatively open in $\overline{\Omega}$ and

$F(D^{2}\varphi(x),D\varphi(x),\varphi(x),x)\geq
f(x)+\epsilon\quad\textrm{a.e. in}\ \mathcal{O}$

and

$\beta\cdot D\varphi(x)\geq\epsilon\quad\textrm{a.e. on}\
\mathcal{O}\cap\partial\Omega,$

$u-\varphi$ cannot attain a local minimum in $\mathcal{O}$.

We remark that it is also possible to take a $C^{2}$-function as a test
function if $F$ is continuous in each variable. In this case, the solution is
called a $C$-viscosity solution. For a further discussion of a $C$-viscosity
solution, we refer the reader to [CC95].

Next we give some tools to treat viscosity solutions. To do this, we introduce
Pucci extremal operators.

######  Definition 2.2.

Let $0<\lambda\leq\Lambda$. For any $M\in S(n)$, the Pucci extremal operator
$\mathcal{M}^{+}$ and $\mathcal{M}^{-}$ are defined as follows:

$\mathcal{M}^{+}(\lambda,\Lambda,X)=\Lambda\sum_{e_{i}>0}e_{i}+\lambda\sum_{e_{i}<0}e_{i}$

and

$\mathcal{M}^{-}(\lambda,\Lambda,X)=\lambda\sum_{e_{i}>0}e_{i}+\Lambda\sum_{e_{i}<0}e_{i},$

where $e_{i}$ are eigenvalues of $X$. Moreover, for $b>0$, we write

$L_{b}^{\pm}(\lambda,\Lambda,u)=\mathcal{M}^{\pm}(\lambda,\Lambda,D^{2}u)\pm
b|Du|,$

respectively.

This definition allows us to introduce the class $S$. These classes can be
considered as classes of viscosity solutions.

######  Definition 2.3.

Let $0<\lambda\leq\Lambda$. We define the class
$\underline{S}(\lambda,\Lambda,b,f)$ ($\overline{S}(\lambda,\Lambda,b,f)$,
respectively) consisting of all functions $u$ such that

$L_{b}^{+}(\lambda,\Lambda,u)\geq
f\qquad\big{(}L_{b}^{-}(\lambda,\Lambda,u)\leq f,\textrm{
respectively}\big{)}$

in the viscosity sense in $\Omega$. We also define

$S(\lambda,\Lambda,b,f)=\overline{S}(\lambda,\Lambda,b,f)\cap\underline{S}(\lambda,\Lambda,b,f)$

and

$S^{\ast}(\lambda,\Lambda,b,f)=\overline{S}(\lambda,\Lambda,b,|f|)\cap\underline{S}(\lambda,\Lambda,b,-|f|).$

######  Remark 2.4.

Let $u$ be a viscosity subsolution (supersolution, respectively) of

$F(D^{2}u,Du,u,x)=f\qquad\textrm{in}\ \Omega,$

where $F=F(X,q,r,x)$ is uniformly elliptic with constants $\lambda,\Lambda$
satisfying the structure condition (1.7). Then we can observe that $u$
satisfies

$L_{b}^{+}(\lambda,\Lambda,u)+F(0,0,u,x)\geq f\qquad\textrm{in}\ \Omega$
$\big{(}L_{b}^{-}(\lambda,\Lambda,u)+F(0,0,u,x)\leq f\qquad\textrm{in}\
\Omega\textrm{, respectively}\big{)}$

in the viscosity sense.

##  3\. Oblique derivative problems

Before establishing $W^{2,p}$-regularity ($p>n$) for the obstacle problem
(1.5), we first discuss some issues concerning the existence, uniqueness and
regularity for the oblique derivative problem (2.3).

We recall Definition 2.3 to start with an Alexandroff-Bakelman-Pucci (ABP)
maximum principle for the oblique boundary problem. See [LZ18, Theorem 2.1]
for the proof.

######  Lemma 3.1.

Let $u$ satisfy

$\displaystyle\left\\{\begin{array}[]{ll}u\in\overline{S}(\lambda,\Lambda,b,f)&\textrm{in
$\Omega,$}\\\ \beta\cdot Du\leq g&\textrm{on $\Gamma\subset\partial\Omega$
}\\\ \end{array}\right.$

in the viscosity sense for $f\in L^{n}(\Omega)$ and $g\in L^{\infty}(\Gamma)$.
Suppose that there exist $\xi\in\partial B_{1}$ and $\delta_{1}>0$ such that
$\beta(x)\cdot\xi\geq\delta_{1}$ for any $x\in\Gamma$. Then

$\sup_{\Omega}u^{-}\leq\sup_{\partial\Omega\backslash\Gamma}u^{-}+C(||f^{+}||_{L^{n}(\Omega)}+\max_{\Gamma}g^{+}),$

where $C$ only depends on $n,\lambda,\Lambda,b,\delta_{1}$ and
$\operatorname{diam}(\Omega)$.

We also give a weak Harnack’s inequality for supersolutions. The proof can be
found in [Win09, Proposition 1.8].

######  Lemma 3.2.

Let $p>n$ and $f\in L^{p}(B_{1})$. Suppose that
$u\in\overline{S}(\lambda,\Lambda,b,f)$ in the viscosity sense and $u\geq 0$
in $B_{1}$. Then there exist $p_{0},C>0$ depending only on $n,\lambda,\Lambda$
and $b$ such that

$||u||_{L^{p_{0}}(B_{1/2})}\leq C(\inf_{B_{1/2}}u+||f||_{L^{p}(B_{1})}).$

The following stability lemma, which can be found in [Win09, Proposition 1.5],
will be used later (see also [CCKS96, Theorem 3.8]).

######  Proposition 3.3.

For $k\in\mathbb{N}$, let $\Omega_{k}\subset\Omega_{k+1}$ be an increasing
sequence of bounded domains and $\Omega:=\cup_{k\geq 1}\Omega_{k}$. Let $F$
and $F_{k}$ be measurable in $x$ and satisfy the structure condition (1.7).
Assume that for $p>n$, $f\in L^{p}(\Omega)$ and $f_{k}\in L^{p}(\Omega_{k})$,
and that $u_{k}\in C(\Omega_{k})$ are $L^{p}$-viscosity subsolutions
(supersolutions, respectively) of $F_{k}(D^{2}u_{k},Du_{k},u_{k},x)=f_{k}$ in
$\Omega_{k}$. Suppose that $u_{k}\to u$ locally uniformly in $\Omega$, and for
$B_{r}(x_{0})\subset\Omega$ and $\varphi\in W^{2,p}(B_{r}(x_{0}))$

$\displaystyle||(s-s_{k})^{+}||_{L^{p}(B_{r}(x_{0}))}\to
0\qquad\big{(}||(s-s_{k})^{-}||_{L^{p}(B_{r}(x_{0}))}\to 0\big{)},$ (3.1)

where $s(x)=F(D^{2}\varphi,D\varphi,u,x)-f(x)$ and
$s_{k}(x)=F(D^{2}\varphi_{k},D\varphi_{k},u_{k},x)-f_{k}(x)$. Then $u$ is an
$L^{p}$-viscosity subsolution (supersolution) of

$F(D^{2}u,Du,u,x)=f(x)\ \textrm{ in}\ \Omega.$

Moreover, if $F$ and $f$ are continuous, then $u$ is a $C$-viscosity
subsolution (supersolution) provided that (3.1) holds for $\varphi\in
C^{2}(B_{r}(x_{0}))$.

Now we return to (2.3). Here we assume that there is a continuous increasing
function $\omega$, defined on $[0,\infty)$ with $\omega(0)=0$, such that

$\displaystyle
F(X_{1},q,r,x_{1})-F(X_{2},q,r,x_{2})\leq\omega(|x_{1}-x_{2}|(|q|+1)+\alpha|x_{1}-x_{2}|^{2})$
(3.2)

holds for any $x_{1},x_{2}\in\Omega$, $q\in\mathbb{R}^{n}$, $r\in\mathbb{R}$,
$\alpha>0$ and $X_{1},X_{2}\in S(n)$ satisfying

$\displaystyle-3\alpha\left(\begin{array}[]{c c}I&0\\\
0&I\end{array}\right)\leq\left(\begin{array}[]{c c}X_{2}&0\\\
0&-X_{1}\end{array}\right)\leq 3\alpha\left(\begin{array}[]{c c}I&-I\\\
-I&I\end{array}\right).$ (3.9)

One can find the following existence and uniqueness for the problem (2.3) in
[Lie13, Theorem 7.19]. We remark that the condition (3.2) is needed to ensure
this lemma.

######  Lemma 3.4.

Assume that $F=F(X,q,r,x)$ is convex in $X$ and continuous in $x$, satisfies
(1.6)-(1.7) and (3.2), and $F(0,0,0,x)\equiv 0$, $\partial\Omega\in C^{3}$,
$f\in L^{p}(\Omega)\cap C(\overline{\Omega})$ for $p>n$, $\beta\in
C^{2}(\partial\Omega)$ with $\beta\cdot\mathbf{n}\geq\delta_{0}$ for some
$\delta_{0}>0$. Then there exists a unique viscosity solution $u$ of (2.3).

We now have the following $W^{2,p}$-estimate for the viscosity solution to
(2.3).

######  Lemma 3.5.

Under the assumptions and conclusion in Lemma 3.4, there exists a small
$\epsilon=\epsilon(n,\lambda,\Lambda,p,\delta_{0})$ such that if

$\displaystyle\sup_{x_{0}\in\overline{\Omega},0<\rho<\rho_{0}}\bigg{(}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{\rho}(x_{0})\cap\Omega}}\Theta_{F}(x,x_{0})^{n}\
dx\bigg{)}^{1/n}\leq\epsilon$ (3.10)

for some $\rho_{0}>0$, then the unique solution $u$ belongs to
$W^{2,p}(\Omega)$ with the following estimate

$\displaystyle||u||_{W^{2,p}(\Omega)}\leq C||f||_{L^{p}(\Omega)}$ (3.11)

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.

###### Proof.

According to [BH20, Theorem 4.6], $u$ belongs to $W^{2,p}(\Omega)$ with

$\displaystyle||u||_{W^{2,p}(\Omega)}\leq
C(||u||_{L^{\infty}(\Omega)}+||f||_{L^{p}(\Omega)})$ (3.12)

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.

Therefore, it suffices to obtain the estimate (3.11). To prove this, we argue
by contradiction. Suppose not. Then there exist sequences $\\{u_{k}\\}$ and
$\\{f_{k}\\}$ such that $u_{k}$ is the viscosity solution of

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u_{k},Du_{k},u_{k},x)=f_{k}&\textrm{in
$\Omega,$}\\\ \beta\cdot Du_{k}=0&\textrm{on $\partial\Omega$}\\\
\end{array}\right.$ (3.15)

with

$\displaystyle||u_{k}||_{W^{2,p}(\Omega)}>k||f_{k}||_{L^{p}(\Omega)}\qquad\textrm{for
each }k\geq 1.$ (3.16)

Consider $\tilde{u}_{k}=t_{k}^{-1}u_{k}$, $\tilde{f}_{k}=t_{k}^{-1}f_{k}$ and
$\tilde{F}_{k}(X,q,r,x)=t_{k}^{-1}F(t_{k}X,t_{k}q,t_{k}r,x),$ where
$t_{k}=||u_{k}||_{W^{2,p}(\Omega)}$. Then $\tilde{u}_{k}$ is a viscosity
solution of

$\displaystyle\left\\{\begin{array}[]{ll}\tilde{F}_{k}(D^{2}\tilde{u}_{k},D\tilde{u}_{k},\tilde{u}_{k},x)=\tilde{f}_{k}&\textrm{in
$\Omega,$}\\\ \beta\cdot D\tilde{u}_{k}=0&\textrm{on $\partial\Omega$.}\\\
\end{array}\right.$

We also see that $\tilde{F}_{k}$ satisfies the assumptions of Lemma 3.4 and
$||\tilde{u}_{k}||_{W^{2,p}(\Omega)}=1$. By (3.16),
$||\tilde{f}_{k}||_{L^{p}(\Omega)}<1/k$ and this tends to zero as
$k\to\infty$. Moreover, by weak compactness theorem, we can extract a proper
subsequence $\\{\tilde{u}_{k_{j}}\\}\subset\\{\tilde{u}_{k}\\}$ such that

$\displaystyle\left\\{\begin{array}[]{ll}\tilde{u}_{k_{j}}\rightharpoonup\tilde{v}&\textrm{in
$W^{2,p}(\Omega),$}\\\ \tilde{u}_{k_{j}}\to\tilde{v}&\textrm{in
$W^{1,p}(\Omega)$}\\\ \end{array}\right.$

for some $\tilde{v}\in W^{2,p}(\Omega)$. Since $p>n$, we also observe that
$W^{1,p}(\Omega)\subset\subset C(\overline{\Omega})$ and this yields
$\tilde{u}_{k_{j}}\to\tilde{v}$ in $C(\overline{\Omega})$. Moreover, we also
observe that for each $j$, $\tilde{u}_{k_{j}}\in
C^{0,\alpha_{0}}(\partial\Omega)$ for some $0<\alpha_{0}<1-n/p$ and
$||D\tilde{u}_{k_{j}}||_{L^{\infty}(\partial\Omega)}\leq C$ for some
$C=C(n,p,\Omega)>0$. Then, from Arzelá-Ascoli criterion, we get

$D\tilde{u}_{k_{j}}\to D\tilde{v}\qquad\textrm{on}\ \partial\Omega.$

Hence, by using Proposition 3.3, we see that $v$ is a viscosity solution of

$\displaystyle\left\\{\begin{array}[]{ll}\tilde{F}(D^{2}\tilde{v},D\tilde{v},\tilde{v},x)=0&\textrm{in
$\Omega,$}\\\ \beta\cdot D\tilde{v}=0&\textrm{on $\partial\Omega$}\\\
\end{array}\right.$ (3.19)

for some $\tilde{F}=\tilde{F}(X,q,r,x)$. Here we can check that $\tilde{F}$
also satisfies (1.6)-(1.7) and (3.2).

Now we deduce that $\tilde{v}\equiv 0$ solves (3.19). Then $\tilde{v}$ is the
unique solution of (3.19) by Lemma 3.4. But, in this case, we get

$\displaystyle 1=||\tilde{u}_{k_{j}}||_{W^{2,p}(\Omega)}\leq
C(||\tilde{u}_{k_{j}}||_{L^{\infty}(\Omega)}+||\tilde{f}_{k_{j}}||_{L^{p}(\Omega)})\to
0\qquad\textrm{as}\ j\to\infty,$

which is a contradiction. ∎

In the above lemma, we have assumed (3.2), which says that $F$ and $f$ are at
least continuous in $x$. By using mollification, we can relax this assumption.

######  Lemma 3.6.

Assume that $F=F(X,q,r,x)$ is convex in $X$ and measurable in $x$, satisfies
(1.6)-(1.7) and $F(0,0,0,x)\equiv 0$, $\partial\Omega\in C^{3}$, $f\in
L^{p}(\Omega)$ for $p>n$, $\beta\in C^{2}(\partial\Omega)$ with
$\beta\cdot\mathbf{n}\geq\delta_{0}$ for some $\delta_{0}>0$. Then there
exists a unique viscosity solution $u$ of (2.3) with the estimate (3.11).

###### Proof.

Fix $\epsilon>0$. With a standard mollifier $\varphi$ having
$\operatorname{supp}\varphi\subset B_{1}$, we define
$\varphi_{\epsilon}(x)=\epsilon^{-n}\varphi(x/\epsilon)$. Then we set
$f^{\epsilon}(x)=(f\ast\varphi_{\epsilon})(x)$ and

$F^{\epsilon}(X,q,r,x)=(F(X,q,r,\cdot)\ast\varphi_{\epsilon})(x).$

Note that we extended $F$ and $f$ to zero outside $\Omega$ here. Then one can
check that $f^{\epsilon}\in L^{p}(\mathbb{R}^{n})\cap
C^{\infty}(\mathbb{R}^{n})$ and $F^{\epsilon}$ is convex in $X$ and
$F^{\epsilon}(0,0,0,x)\equiv 0$. Furthermore, we also observe that
$F^{\epsilon}$ satisfies (1.6)-(1.7), (3.2) and (3.10) (see the proof of
[Win09, Theorem 4.3]). Consider the following problem

$\displaystyle\left\\{\begin{array}[]{ll}F^{\epsilon}(D^{2}u_{\epsilon},Du_{\epsilon},u_{\epsilon},x)=f^{\epsilon}&\textrm{in
$\Omega,$}\\\ \beta\cdot Du_{\epsilon}=0&\textrm{on $\partial\Omega$.}\\\
\end{array}\right.$ (3.22)

Then applying Lemma (3.5) to $u_{\epsilon}$, there exists the unique solution
$u_{\epsilon}$ of (3.22) with

$\displaystyle||u_{\epsilon}||_{W^{2,p}(\Omega)}\leq
C||f^{\epsilon}||_{L^{p}(\Omega)}$

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.

Since $W^{2,p}(\Omega)\subset\subset C^{1,\alpha}(\overline{\Omega})$ with
$0<\alpha<1-n/p$ by Sobolev imbedding, we have
$\\{u_{\epsilon}\\}_{\epsilon>0}$ is uniformly bounded in
$C^{1,\alpha_{0}}(\overline{\Omega})$ for any small $\epsilon>0$ and some
$0<\alpha_{0}<1-n/p$. Thus, by using Arzelá-Ascoli criterion, we can obtain
that there exists a function $v$ with

$u_{\epsilon_{j}}\to v\qquad\textrm{uniformly in}\ \overline{\Omega}$

for some subsequence $\\{u_{\epsilon_{j}}\\}\subset\\{u_{\epsilon}\\}$. Again,
applying Proposition 3.3 to $v$, we can derive that $v$ is a viscosity
solution of (2.3) with

$\displaystyle||v||_{W^{2,p}(\Omega)}\leq C||f||_{L^{p}(\Omega)}$

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.
The uniqueness can be deduced by Lemma 3.7 below. ∎

Meanwhile, a comparison principle for (2.3) can be also obtained as in the
case of Dirichlet problems, see [CCKS96, Theorem 2.10]. This will be used to
prove Theorem 1.1 in the next section.

######  Lemma 3.7.

Let $\Omega_{0}\subset\Omega$, and let $\Gamma\in C^{2}$ be relatively open in
$\partial\Omega$. Suppose that $F=F(X,q,r,x)$ is convex in $X$ and continuous
in $x$, satisfies (1.6)-(3.2) and $F(0,0,0,x)\equiv 0$, $\beta\in
C^{2}(\overline{\Gamma})$ with $\beta\cdot\mathbf{n}\geq\delta_{0}$ for some
$\delta_{0}>0$, $\psi\in C(\partial\Omega_{0}\backslash\Gamma)$ and $f\in
L^{p}(\Omega_{0})$ for $n<p<\infty$.

Let $u_{1},u_{2}\in W^{2,p}(\Omega)\cap C(\overline{\Omega})$ satisfy

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u_{1},Du_{1},u_{1},x)\leq
f&\textrm{in $\Omega_{0},$}\\\ u_{1}\geq\psi&\textrm{on
$\partial\Omega_{0}\backslash\Gamma$,}\\\ \beta\cdot Du_{1}\leq 0&\textrm{on
$\Gamma$}\\\ \end{array}\right.$

and

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u_{2},Du_{2},u_{2},x)\geq
f&\textrm{in $\Omega_{0},$}\\\ u_{2}\leq\psi&\textrm{on
$\partial\Omega_{0}\backslash\Gamma$,}\\\ \beta\cdot Du_{2}\geq 0&\textrm{on
$\Gamma$ }\\\ \end{array}\right.$

in the viscosity sense. Then we have $u_{1}\geq u_{2}$ in $\Omega_{0}$.

###### Proof.

First we set

$G(X,q,r,x)=F(X+D^{2}u_{2},q+Du_{2},r+u_{2},x)-F(D^{2}u_{2},Du_{2},u_{2},x).$

One can see that $G$ satisfies (1.6)-(3.2), $w:=u_{1}-u_{2}$ solves

$\displaystyle
G(D^{2}w,Dw,w,x)=F(D^{2}u_{1},Du_{1},u_{1},x)-F(D^{2}u_{2},Du_{2},u_{2},x)\leq
0$

in the viscosity sense, and that $w\in\overline{S}(\lambda,\Lambda,b,-g)$ for

$g(x)=G(0,0,w,x)=F(D^{2}u_{2},Du_{2},u_{1},x)-F(D^{2}u_{2},Du_{2},u_{2},x).$

Set

$V:=\\{x\in\overline{\Omega}_{0}:w(x)<0\\}=\\{x\in\overline{\Omega}_{0}:u_{1}(x)<u_{2}(x)\\}.$

We want to claim that $V=\varnothing$. Suppose not. Then we have
$\inf_{V}w<0$. Since

$g(x)>d(u_{2}-u_{1})(x)=-dw(x)>0\qquad\textrm{for}\ x\in V,$

$w$ satisfies

$\displaystyle\left\\{\begin{array}[]{ll}w\in\overline{S}(\lambda,\Lambda,b,0)&\textrm{in
$V,$}\\\ w\geq 0&\textrm{on $\partial V\backslash\Gamma$,}\\\ \beta\cdot
Dw\leq 0&\textrm{on $\Gamma$}\\\ \end{array}\right.$

in the viscosity sense, according to [LZ18, Theorem 3.1].

We first consider the case $\partial V\backslash\Gamma\neq\varnothing$.
Observe that $w\equiv 0$ on $\partial V\cap\Omega_{0}$ and $w\geq 0$ on
$(\partial V\cap\partial\Omega_{0})\backslash\Gamma$. From ABP maximum
principle (see [CCKS96, Proposition 3.3]), we can deduce that

$\sup_{V}w^{-}=\sup_{\partial V}w^{-}.$

Thus, $w$ attains a minimum point $x_{0}$ on $\partial V\cap\Gamma$.

Now define $\tilde{w}(x)=w(x)-\inf_{V}w$ for $x\in V$. Then we see that
$\tilde{w}\geq 0$ in $\overline{V}$ with $\tilde{w}(x_{0})=0$. Consider a
small neighborhood $N(x_{0})\subset\Omega\cup\Gamma$ of $x_{0}$. According to
[CMS11, Proposition 11], we can deduce that there exists a point $x_{1}\in
N(x_{0})\cap\Omega$ with $\tilde{w}(x_{1})=0$ since $\beta\cdot Dw\leq 0$.
Then, by Lemma 3.2, there exists a small number $\rho>0$ with
$B_{\rho}(x_{1})\subset\subset\Omega$ such that

$||\tilde{w}||_{L^{p_{0}}(B_{\rho/2}(x_{1}))}=0.$

That is,

$\tilde{w}\equiv 0\qquad\textrm{in}\quad B_{\rho/2}(x_{1}).$

Repeating the above procedure, we can deduce that

$\tilde{w}\equiv 0\qquad\textrm{in}\quad V.$

Since $\tilde{w}$ is continuous in $\overline{V}$, we have

$\tilde{w}\equiv 0\qquad\textrm{in}\quad\overline{V},$

and this implies $w=\tilde{w}$. However, it is a contradiction, as we have
assumed that $\inf_{V}w<0.$ Hence, $V=\varnothing$ and we conclude that

$w\geq 0\qquad\textrm{in}\quad\Omega_{0},$

and this implies

$u_{1}\geq u_{2}\qquad\textrm{in}\quad\Omega_{0}.$

On the other hand, if $\Gamma=\partial V$, we have

$\displaystyle\left\\{\begin{array}[]{ll}w\in\overline{S}(\lambda,\Lambda,b,0)&\textrm{in
$V,$}\\\ \beta\cdot Dw\leq 0&\textrm{on $\Gamma$.}\\\ \end{array}\right.$

Again, by ABP maximum principle, we also have

$\sup_{V}w^{-}=\sup_{\partial V}w^{-}<0.$

Set $\tilde{w}=w-\inf_{V}w$ in $V$. Then there is a point $y_{0}\in\partial V$
with $\tilde{w}(y_{0})=0$. From Lemma 3.1, we can see that there is an
interior point $y_{1}$ such that $\tilde{w}(y_{1})=0$. Now we derive that
$\tilde{w}\equiv 0$ in V by using a similar argument as above. This yields
that $w\equiv c_{1}$ in $V$ for some $c_{1}<0$.

By the definition of $w$, we have $u_{1}\equiv u_{2}+c_{1}$ in $V$. Then we
can observe that

$\displaystyle F(D^{2}u_{1},Du_{1},u_{1},x)$
$\displaystyle=F(D^{2}(u_{2}+c_{1}),D(u_{2}+c_{1}),u_{2}+c_{1},x)$
$\displaystyle=F(D^{2}u_{2},Du_{2},u_{2}+c_{1},x)$ $\displaystyle\geq
F(D^{2}u_{2},Du_{2},u_{2},x)-dc_{1}$ $\displaystyle>f$

in $V$. But it is a contradiction because $F(D^{2}u_{1},Du_{1},u_{1},x)\leq
f$. Therefore, we conclude that $V=\varnothing$. This completes the proof. ∎

##  4\. Proof of Theorem 1.1

In this section we establish our main result, Theorem 1.1. Our strategy is to
construct a sequence of approximating oblique derivative problems to (1.5).
This construction makes it possible to utilize those results obtained in the
previous section for (2.3).

In the process of the proof, we are going to use the following Schauder’s
fixed point theorem (see [Con85, Theorem V.9.5]).

######  Lemma 4.1 (Schauder’s fixed point theorem).

Assume that $X$ is a Banach space, $K\subset X$ is closed, bounded and convex,
and suppose that $S:K\to K$ is compact. Then $S$ has a fixed point in $K$.

We now prove the main result of this paper.

###### Proof of Theorem 1.1.

Fix $\epsilon>0$ and choose a non-decreasing function $\Phi_{\epsilon}\in
C^{\infty}(\mathbb{R})$ such that

$\displaystyle\Phi_{\epsilon}(s)\equiv 0\quad\textrm{if}\quad s\leq
0;\qquad\Phi_{\epsilon}(s)\equiv 1\quad\textrm{if}\quad s\geq\epsilon,$ (4.1)

and

$\displaystyle 0\leq\Phi_{\epsilon}(s)\leq 1\quad\textrm{for any}\
\,s\in\mathbb{R}.$ (4.2)

Set

$g(x):=f(x)-F(D^{2}\psi,D\psi,\psi,x).$

Then we have $g\in L^{p}(\Omega)$ with the estimate

$\displaystyle\begin{split}||g||_{L^{p}(\Omega)}&\leq||f||_{L^{p}(\Omega)}+||F(D^{2}\psi,D\psi,\psi,x)||_{L^{p}(\Omega)}\\\
&\leq C(||f||_{L^{p}(\Omega)}+||\psi||_{W^{2,p}(\Omega)})\end{split}$ (4.3)

for some $C=C(n,\lambda,\Lambda,b,c)>0$, since $f,D^{2}\psi\in L^{p}(\Omega)$
and $F$ satisfies (1.7).

We now consider the following oblique derivative problem without obstacles

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u_{\epsilon},Du_{\epsilon},u_{\epsilon},x)=g^{+}\Phi_{\epsilon}(u_{\epsilon}-\psi)+f-g^{+}&\textrm{in
$\Omega,$}\\\ \beta\cdot Du_{\epsilon}=0&\textrm{on $\partial\Omega$.}\\\
\end{array}\right.$ (4.6)

We want to show that (4.6) has a unique viscosity solution. For this, fix a
function $v_{0}\in L^{p}(\Omega)$. Then according to Lemma 3.6, we know that
there exists a unique viscosity solution $v_{\epsilon}\in W^{2,p}(\Omega)$ of

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}v_{\epsilon},Dv_{\epsilon},v_{\epsilon},x)=g^{+}\Phi_{\epsilon}(v_{0}-\psi)+f-g^{+}&\textrm{in
$\Omega,$}\\\ \beta\cdot Dv_{\epsilon}=0&\textrm{on
$\partial\Omega$}\end{array}\right.$

with the estimate

$\displaystyle||v_{\epsilon}||_{W^{2,p}(\Omega)}$ $\displaystyle\leq
C||g^{+}\Phi_{\epsilon}(v_{0}-\psi)+f-g^{+}||_{L^{p}(\Omega)}$
$\displaystyle\leq C(||f||_{L^{p}(\Omega)}+||g||_{L^{p}(\Omega)})$
$\displaystyle\leq C(||f||_{L^{p}(\Omega)}+||\psi||_{W^{2,p}(\Omega)})$

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\rho_{0},\operatorname{diam}(\Omega))>0$,
where we have used (4.1) and (4.2). Thus,

$||v_{\epsilon}||_{W^{2,p}(\Omega)}\leq C_{0}$

for some

$C_{0}=C_{0}(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)}\operatorname{diam}(\Omega),||f||_{L^{p}(\Omega)},||\psi||_{L^{p}(\Omega)},\rho_{0}).$

Note that $C_{0}$ is independent of $v_{0}$. Now we can define a nonlinear
operator $S_{\epsilon}:L^{p}(\Omega)\to W^{2,p}(\Omega)\subset L^{p}(\Omega)$
such that $S_{\epsilon}v_{0}=v_{\epsilon}$ with (4.6). Write

$K:=\\{h\in L^{p}(\Omega):||h||_{L^{p}(\Omega)}\leq C_{0}\\}.$

Note that $K$ is a closed convex subset of $L^{p}(\Omega)$. On the other hand,
by Rellich-Kondrachov compactness theorem, we observe that $W^{2,p}(\Omega)$
is compactly imbedded in $W^{1,p}(\Omega)$ and so is in $L^{p}(\Omega)$.
Hence, the closure of $S_{\epsilon}(A)$ is compact for every $A\subset K$.
Meanwhile, by Proposition 3.3, we can also conclude that $S_{\epsilon}$ is a
continuous operator.

Therefore, by Lemma 4.1, there exists a function $u_{\epsilon}\in K$
satisfying $S_{\epsilon}u_{\epsilon}=u_{\epsilon}$, and this implies that
$u_{\epsilon}$ is a viscosity solution of (4.6). Furthermore, from (4.3) and
Lemma 3.6, we also observe that

$\displaystyle||u_{\epsilon}||_{W^{2,p}(\Omega)}\leq
C(||f||_{L^{p}(\Omega)}+||\psi||_{W^{2,p}(\Omega)})$ (4.7)

for some
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.
This shows that $\\{u_{\epsilon}\\}_{\epsilon>0}$ is uniformly bounded in
$W^{2,p}(\Omega)$.

Recall that $p>n$. Then we observe that $W^{2,p}(\Omega)\subset\subset
C^{1,\alpha_{0}}(\overline{\Omega})$ for some $0<\alpha_{0}<1-n/p$ by Morrey
imbedding. Therefore, we can find a subsequence $\\{u_{\epsilon_{j}}\\}$ with
$\epsilon_{j}\searrow 0$ and a function $u\in W^{2,p}(\Omega)$ such that

$\displaystyle\left\\{\begin{array}[]{ll}u_{\epsilon_{j}}\rightharpoonup
u\quad\textrm{in}\ W^{2,p}(\Omega),\\\ u_{\epsilon_{j}}\to u\quad\textrm{in}\
W^{1,p}(\Omega)\subset C^{0,\alpha_{0}}(\overline{\Omega})\\\
\end{array}\right.$ (4.10)

as $j\to\infty$.

Now we claim that $u$ is indeed the unique viscosity solution of (1.5). We
first see that $u$ is uniformly bounded and equicontiuous on $\partial\Omega$
from (4.7) and Morrey imbedding. Thus, by using Arzelá-Ascoli criterion, we
have

$\beta\cdot Du=0\quad\textrm{on}\ \partial\Omega$

in the viscosity sense. On the other hand, from (4.6), we observe that

$F(D^{2}u_{\epsilon_{j}},Du_{\epsilon_{j}},u_{\epsilon_{j}},x)=g^{+}\Phi_{\epsilon_{j}}(u_{\epsilon_{j}}-\psi)+f-g^{+}\leq
f\qquad\textrm{in}\ \Omega$

for each $j$. Passing to the limit $j\to\infty$, we have $F(D^{2}u,Du,u,x)\leq
f$ in $\Omega$.

Next, we show that

$u\geq\psi\quad\textrm{in}\ \overline{\Omega}.$

We first see that $\Phi_{\epsilon_{j}}(u_{\epsilon_{j}}-\psi)\equiv 0$ on the
set

$V_{j}=\\{x\in\overline{\Omega}:u_{\epsilon_{j}}(x)<\psi(x)\\}.$

If $V_{j}=\varnothing$, we have $u_{\epsilon_{j}}\geq\psi$ in
$\overline{\Omega}$, and so we are done. Now suppose that
$V_{j}\neq\varnothing$. Then,

$F(D^{2}u_{\epsilon_{j}},Du_{\epsilon_{j}},u_{\epsilon_{j}},x)=f(x)-g^{+}(x)\qquad\textrm{for}\
x\in V_{j}.$

We note that $V_{j}$ is relatively open in $\overline{\Omega}$ for each $j$
since $u_{\epsilon_{j}}\in C(\overline{\Omega})$.

Recall that

$F(D^{2}\psi,D\psi,\psi,x)=f-g\geq
F(D^{2}u_{\epsilon_{j}},Du_{\epsilon_{j}},u_{\epsilon_{j}},x)\quad\textrm{in}\
V_{j}.$

And we also have $u_{\epsilon_{j}}=\psi$ on $\partial
V_{j}\backslash\partial\Omega$.

Now we can apply Lemma 3.7 to obtain $u_{\epsilon_{j}}\geq\psi$ in $V_{j}$,
which is a contradiction to the definition of $V_{j}$ and thus
$V_{j}=\varnothing$ for each $j$. Therefore, we can obtain $u\geq\psi$ in
$\overline{\Omega}$.

We next claim that

$F(D^{2}u,Du,u,x)=f\qquad\textrm{in}\ V:=\\{x\in\Omega:u(x)>\psi(x)\\}.$

For each $m\in\mathbb{N}$, we have

$\Phi_{\epsilon_{j}}(u_{\epsilon_{j}}-\psi)\to 1\quad\textrm{a.e. in}\
\bigg{\\{}x\in\Omega:u(x)>\psi(x)+\frac{1}{m}\bigg{\\}}$

as $j\to\infty$. Thus, for

$V=\\{x\in\Omega:u(x)>\psi(x)\\}=\bigcup_{m=1}^{\infty}\bigg{\\{}x\in\Omega:u(x)>\psi(x)+\frac{1}{m}\bigg{\\}},$

we derive

$g^{+}\Phi_{\epsilon_{j}}(u_{\epsilon_{j}}-\psi)+f-g^{+}\to
f\qquad\textrm{a.e.}\ \ \textrm{in}\ V$

as $j\to\infty$. Thus, we deduce that

$F(D^{2}u,Du,u,x)=g^{+}+f-g^{+}=f\qquad\textrm{in}\ V$

in the viscosity sense.

Therefore, we can conclude that $u$ is a viscosity solution of (1.5).
Moreover, from (4.7) and (4.10), we have

$\displaystyle||u||_{W^{2,p}(\Omega)}\leq\liminf_{j\to\infty}||u_{\epsilon_{j}}||_{W^{2,p}(\Omega)}\leq
C(||f||_{L^{p}(\Omega)}+||\psi||_{W^{2,p}(\Omega)})$

for some constant
$C=C(n,\lambda,\Lambda,p,\delta_{0},b,c,||\beta||_{C^{2}(\partial\Omega)},\operatorname{diam}(\Omega),\rho_{0})$.

For the uniqueness, let $u_{1}$ and $u_{2}$ be two viscosity solutions of
(1.5). Suppose that $u_{1}\not\equiv u_{2}$. Then we can assume without loss
of generality that

$G=\\{u_{2}>u_{1}\\}\neq\varnothing.$

Since $u_{2}>u_{1}\geq\psi$ in $G$, we see that
$F(D^{2}u_{2},Du_{2},u_{2},x)=f$ in $G$ in the viscosity sense. Then we have

$\displaystyle\left\\{\begin{array}[]{ll}F(D^{2}u_{1},Du_{1},u_{1},x)\leq
F(D^{2}u_{2},Du_{2},u_{2},x)=f&\textrm{in $G,$}\\\ u_{1}=u_{2}&\textrm{on
$\partial G\backslash\partial\Omega$,}\\\ \beta\cdot Du_{1}=\beta\cdot
Du_{2}=0&\textrm{on $\partial G\cap\partial\Omega$.}\\\ \end{array}\right.$

Now applying [CCKS96, Theorem 2.10] or Lemma 3.7 to $u_{1}-u_{2}$, we deduce
that $u_{1}\geq u_{2}$ in $G$ whether $\partial
G\cap\partial\Omega=\varnothing$ or not. This contradicts the definition of
the set $G$, and hence $u_{1}=u_{2}$. ∎

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