Regularity for fully nonlinear elliptic equations with oblique boundary conditions

TL;DR

This paper establishes regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique boundary conditions, including pointwise $C^{eta}$, $C^{1,eta}$, and $C^{2,eta}$ regularity, along with fundamental principles like maximum principle and Harnack inequality.

Contribution

It provides new regularity results and fundamental principles for solutions of fully nonlinear elliptic equations with oblique boundary conditions.

Findings

Proved pointwise $C^{eta}$, $C^{1,eta}$, and $C^{2,eta}$ regularity.

Established the A-B-P maximum principle and Harnack inequality.

Demonstrated uniqueness and solvability of the equations.

Abstract

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^{\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.

Equations

$$\{\begin{array}{rlrl}& F(D^{2}u)=f& & \text{mbox}in\Omega ;\\ & \beta \cdot Du+\gamma u=g& & \text{mbox}on\Gamma \subset \partial \Omega ,\end{array}$$

$$\beta \cdot \vec{n}\ge {\delta }_0\text{mbox}on\Gamma \text{mbox}and\Vert \beta {\Vert }_{L^{\infty }(\Gamma )}\le 1$$

$$\lambda \Vert N\Vert \le F(M+N)-F(M)\le \Lambda \Vert N\Vert ,\forall M,N\in S^n,N\ge 0,$$

$$\{\begin{array}{rlrl}& u\in S(\lambda ,\Lambda ,f)& & \text{mbox}in\Omega ;\\ & \beta \cdot Du+\gamma u=g& & \text{mbox}on\Gamma .\end{array}$$

$$\Vert u{\Vert }_{C^{\alpha }(\overline{{\Omega }^{\prime }})}\le C\left(\Vert u{\Vert }_{L^{\infty }(\Omega )}+\Vert f{\Vert }_{L^n(\Omega )}+\Vert g{\Vert }_{L^{\infty }(\Gamma )}\right),$$

$${\left(\frac{1}{|B_r(x_0)\cap \Omega |}{\int }_{B_r(x_0)\cap \Omega }|f{|}^n\right)}^{\frac{1}{n}}\le C_f{r}^{\alpha -1}\forall x_0\in \Omega ,\forall r>0.$$

$$\Vert u{\Vert }_{C^{1,\alpha }(\overline{{\Omega }^{\prime }})}\le C\left(\Vert u{\Vert }_{L^{\infty }(\Omega )}+C_f+\Vert g{\Vert }_{C^{\alpha }(\bar{\Gamma })}+|F(0)|\right),$$

$$\Vert u{\Vert }_{C^{2,\alpha }(\overline{{\Omega }^{\prime }})}\le C\left(\Vert u{\Vert }_{L^{\infty }(\Omega )}+\Vert f{\Vert }_{C^{\alpha }(\bar{\Omega })}+\Vert g{\Vert }_{C^{1,\alpha }(\bar{\Gamma })}+|F(0)|\right),$$

$$F(D^2\phi (x_0))\ge (\le )f(x_0)\text{mbox}if{x}_0\in \Omega$$

$$\beta (x_0)\cdot D\phi (x_0)+\gamma (x_0)\phi (x_0)\ge (\le )g(x_0)\text{mbox}if{x}_0\in \Gamma .$$

$$\{\begin{array}{rlrl}& u_m\in S(\lambda ,\Lambda ,f)& & \text{mbox}in\Omega ;\\ & \beta \cdot D{u}_m+\gamma u_m=g& & \text{mbox}on\Gamma .\end{array}$$

$$\{\begin{array}{rlrl}& u\in S(\lambda ,\Lambda ,f)& & \text{mbox}in\Omega ;\\ & \beta \cdot Du+\gamma u=g& & \text{mbox}on\Gamma .\end{array}$$

$$P_n(0)<\frac{1}{{\beta }_n(0)}\left(g(0)-\gamma (0)P(0)-{\beta }^{\prime }(0)\cdot D_{\_}P(0)\right):=A.$$

$$P_n(x)<A-\eta \forall x\in B_{\tau }.$$

$$P(x)=$$

$$\le$$

$$P(x)\le P(x^{\prime },\phi (x^{\prime }))+(A-\eta /2)(x_n-\phi (x^{\prime }))-N(x_n-\phi (x^{\prime }){)}^2\forall x\in \bar{\Omega }\cap B_{{\tau }^{\prime }}.$$

$${\beta }_n\ge {\delta }_0/2\text{mbox}on\Gamma \cap B_{{\tau }^{\prime }}.$$

$${\psi }_{\epsilon }\ge P\ge u\text{mbox}in\Omega \cap B_{{\tau }^{\prime }}\text{mbox}and{\psi }_{\epsilon }(0)=P(0)=u(0).$$

$$\Vert u_{m_0}-u{\Vert }_{L^{\infty }(\Omega \cap B_{{\tau }^{\prime }})}\le \frac{1}{2}(\epsilon {\tau }^{\prime }{)}^2.$$

$$|{\psi }_{\epsilon }(0)-u_{m_0}(0)|\le \frac{1}{2}(\epsilon {\tau }^{\prime }{)}^2\text{mbox}and{\psi }_{\epsilon }-u_{m_0}>\frac{1}{2}(\epsilon {\tau }^{\prime }{)}^2\text{mbox}on\partial (\Omega \cap B_{{\tau }^{\prime }})\backslash \Gamma .$$

$${\psi }_{\epsilon ,ij}=\{\begin{array}{rlrl}& (-A+\eta /2-2N\phi +2N{x}_n){\phi }_{ij}-2N{\phi }_i{\phi }_j& & \\ & +P_{ij}+P_{in}{\phi }_j+P_{jn}{\phi }_i+P_n{\phi }_{ij}+2\epsilon ,& & i,j<n,\\ & 2N{\phi }_i,& & i<n,j=n,\\ & -2N+2\epsilon ,& & i=j=n.\end{array}$$

$$M^{+}\left(D^2{\psi }_{\epsilon }\right)<-\Vert f{\Vert }_{L^{\infty }(\Omega )}\text{mbox}in\Omega \cap B_{{\tau }^{\prime }}.$$

$$\beta (\tilde{x})\cdot D{\psi }_{\epsilon }(\tilde{x})+\gamma (\tilde{x}){\psi }_{\epsilon }(\tilde{x})\ge g(\tilde{x}).$$

$$D{\psi }_{\epsilon }=(A-\eta /2)(-D\phi ,1)-2\epsilon \tilde{x}+(D_{\_}P+P_{n}D\phi ,0)\text{mbox}at\tilde{x}.$$

$$g\le (\eta /2-A){\beta }^{\prime }D\phi +{\beta }_n(A-\eta /2)-2\epsilon \beta \tilde{x}+{\beta }^{\prime }(D_{\_}P+P_{n}D\phi )+\gamma u.$$

$$g\le {\beta }_n(A-\eta /4)+{\beta }^{\prime }{D}_{\_}P+\gamma u\text{mbox}at\tilde{x}.$$

$$g(\tilde{x})=$$

$$+{\beta }^{\prime }(\tilde{x})\cdot D_{\_}P(\tilde{x})++\gamma (\tilde{x})u(\tilde{x}).$$

$$-{\beta }_n(\tilde{x})\eta /8\ge 0$$

$$\{\begin{array}{rlrl}& u\in \bar{S}(\lambda ,\Lambda ,f)& & \text{mbox}in\Omega ;\\ & \beta \cdot Du\le g& & \text{mbox}on\Gamma .\end{array}$$

Full text

††journal: Archive for Rational Mechanics and Analysis

1 Introduction

We study the regularity of viscosity solutions to the following fully nonlinear elliptic equation with oblique boundary condition:

[TABLE]

where $\Omega\subset R^{n}$ is a bounded domain; $\Gamma\in C^{1}$ and is relatively open to $\partial\Omega$; $f$ is a function defined in $\Omega$; and $\beta$ (vector valued), $\gamma$ and $g$ are functions defined on $\Gamma$. Here “Oblique” means that $|\beta\cdot\vec{n}|\geq\delta_{0}>0$ on $\Gamma$, where $\vec{n}$ denotes the inner normal of $\Omega$. We call $\Gamma$ the oblique boundary. Since the sign of $\gamma$ is not required, without loss of generality, throughout this paper, we always assume that

[TABLE]

for some positive constant $\delta_{0}$. In addition, we always assume that $F$ is uniformly elliptic, i.e., there exist positive constants $\lambda$ and $\Lambda$ such that

[TABLE]

where $S^{n}$ denotes the set of $n\times n$ symmetric matrices; $N\geq 0$ means the nonnegativeness and $\|N\|$ is the spectral radius of $N$. Since we deal with the viscosity solutions of1.1, it is convenient for us to assume that $f$, $\beta$, $\gamma$ and $g$ are continuous throughout this paper.

Few regularity but some a priori estimates and existence results are known for1.1, where $\Gamma=\partial\Omega$ and $\gamma\leq 0$ are needed. For linear equations, the existence of $C^{2,\alpha}$ solutions are obtained by Green’s representation, the method of freezing the coefficients and method of continuity (cf. (G-T, Chapter 6)). The work of Lieberman Li2 covered above results and only required the $C^{1,\alpha}$ smoothness on $\partial\Omega$. Safonov Sa1; Sa2 extended this results to the nonlinear Bellman equations, that is, he proved the existence of $C^{2,\alpha}$ solutions as $\partial\Omega\in C^{1,\alpha}$ for1.1. In 1982, Lieberman Li1 obtained the existence of solutions for quasilinear equations based on a fixed point theorem and the solvability for linear equations. By a priori estimates and the method of continuity, Lieberman and Trudinger L-T proved the existence and uniqueness of $C^{2,\alpha}$ solutions for fully nonlinear elliptic equations with nonlinear oblique boundary conditions as $\partial\Omega\in C^{4}$.

Since 1980s, the notion of viscosity solutions has been applied widely in the study of non-divergent equations especially of fully nonlinear elliptic equations; and some important interior regularity results and global results with Dirichlet boundary conditions have been obtained (see C-C and C-I-L and references therein). As for applications of viscosity solutions to oblique derivative problems, Ishii Is1 obtained the existence and uniqueness of viscosity solutions for fully nonlinear elliptic equations. Milakis and Silvestre M-S proved the $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity results for fully nonlinear equations with Neumann boundary data on flat boundaries. For more a priori estimates of the oblique derivative problems, we refer the reader to the book Li7.

In this paper, we derive a series of regularity results for viscosity solutions of1.1. In particular, the $C^{\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ boundary regularity are deduced:

Theorem 1.1 ($C^{\alpha}$ regularity)

Let $u$ satisfy

[TABLE]

Then for any $\Omega^{\prime}\subset\subset\Omega\cup\Gamma$, $u\in C^{\alpha}(\bar{\Omega^{\prime}})$ and

[TABLE]

where $0<\alpha<1$ depends only on $n$, $\lambda$, $\Lambda$ and $\delta_{0}$, and $C$ depends also on $\|\gamma\|_{L^{\infty}(\Gamma)}$, $\Omega^{\prime}$ and $\Omega$.

Theorem 1.2 ($C^{1,\alpha}$ regularity)

Let $u$ be a viscosity solution of1.1 and $0<\alpha<\bar{\alpha}$ where $0<\bar{\alpha}<1$ is a constant depending only on $n$, $\lambda$, $\Lambda$ and $\delta_{0}$. Suppose that $\beta,\gamma,g\in C^{\alpha}(\bar{\Gamma})$ and $f$ satisfies

[TABLE]

Then for any $\Omega^{\prime}\subset\subset\Omega\cup\Gamma$, $u\in C^{1,\alpha}(\bar{\Omega^{\prime}})$ and

[TABLE]

where $C$ depends only on $n$, $\lambda$, $\Lambda$, $\delta_{0}$, $\alpha$, $\|\beta\|_{C^{\alpha}(\bar{\Gamma})}$, $\|\gamma\|_{C^{\alpha}(\bar{\Gamma})}$, $\Omega^{\prime}$ and $\Omega$.

Theorem 1.3 ($C^{2,\alpha}$ regularity)

Let $F$ be convex, $u$ be a viscosity solution of1.1 and $0<\alpha<\tilde{\alpha}$ where $0<\tilde{\alpha}<1$ is a constant depending only on $n$, $\lambda$, $\Lambda$ and $\delta_{0}$. Suppose that $\Gamma\in C^{1,\alpha}$, $\beta,\gamma,g\in C^{1,\alpha}(\bar{\Gamma})$ and $f\in C^{\alpha}(\bar{\Omega})$.

Then for any $\Omega^{\prime}\subset\subset\Omega\cup\Gamma$, $u\in C^{2,\alpha}(\bar{\Omega^{\prime}})$ and

[TABLE]

where $C$ depends only on $n$, $\lambda$, $\Lambda$, $\delta_{0}$, $\alpha$, $\|\beta\|_{C^{1,\alpha}(\bar{\Gamma})}$, $\|\gamma\|_{C^{1,\alpha}(\bar{\Gamma})}$, $\Omega^{\prime}$and $\Omega$.

The following is the outline of this paper. First, the Alexandrov-Bakel’man-Pucci type maximum principle and boundary Harnack type inequality are presented as basic tools. Then $C^{\alpha}$ regularity follows clearly. The $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity for viscosity solutions of1.1 are obtained by approximating the original problem by model problems at each scales whose regularity can be dealt with and which should be designed properly. We point out here that we do not need flatten the boundary, which is different from the references mentioned above. This paper also contains uniqueness and existence results of1.1, which will be used to solve our model problems.

The paper is organized in the following way. We introduce the Alexandrov-Bakel’man-Pucci (A-B-P for short) maximum principle and the boundary Harnack inequality in Section 2, which are the basic tools to attack the regularity for1.1. Based the boundary Harnack inequality, the pointwise $C^{\alpha}$ regularity follows. The proof for the A-B-P maximum principle has been motivated by M-S, where the authors deal with the Neumann problems. Combining the A-B-P maximum principle, the interior Harnack inequality and the barrier technique, we derive the boundary Harnack inequality. The barrier is adopted from L-T.

A Jensen’s type uniqueness of viscosity solutions is presented in LABEL:S3. That is, the subsolution minus the supersolution is also a “subsolution”. This leads to the uniqueness of viscosity solutions by combining with the A-B-P maximum principle, where a mixed boundary value problem is considered. In addition, we also prove the existence result by Perron’s method, which will be used to construct our model problem to approximate the viscosity solution of1.1 later.

We present our model problem and prove the $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity of its solution in LABEL:S4, which will be used to approximate the viscosity solution of1.1 at different scales. Roughly speaking, the model problem is that $f,\gamma,g\equiv 0$, $\beta$ is a constant vector and $\Omega$ is a spherical cap with $\Gamma$ is the flat part. The $C^{1,\alpha}$ regularity is a consequence of the Jensen’s type uniqueness obtained in LABEL:S3. To show the $C^{2,\alpha}$ regularity under the assumption that $F$ is convex, we first prove that the restriction of the viscosity solution on the flat boundary is also a viscosity solution in dimension $n-1$, and then by the interior $C^{2,\alpha}$ estimates for convex operators, we have the $C^{2,\alpha}$ regularity of the restriction solution. Finally, from the boundary $C^{2,\alpha}$ regularity for the Dirichlet problem, we deduce the conclusion.

Based on the previous results, we show the pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity for1.1 in LABEL:S5 and LABEL:S6 respectively, where higher regularities for1.1 is also presented in LABEL:S6. Our proof is influenced by the spirits in C-C.

Before the end of this section, we introduce the following notations which will be frequently used in this paper.

Notation 1.1

$\{e_{i}\}^{n}_{i=1}$: the standard basis of $R^{n}$, i.e., $e_{i}=(0,...0,\underset{i^{th}}{1},0,...0)$. 2. 2.

Given $x=(x_{1},x_{2},...,x_{n})\in R^{n}$, we may rewrite $x=(x^{\prime},x_{n})$ where $x^{\prime}=(x_{1},x_{2},...,x_{n-1})$ . 3. 3.

$S^{n}$: the set of $n\times n$ symmetric matrices and $\|A\|:=$ the spectral radius of $A$ for any $A\in S^{n}$. 4. 4.

R^{n}_{+}:=\{x\in R^{n}\big{|}x_{n}>0\}. 5. 5.

Given $\beta\in R^{n}$ with $\beta_{n}\neq 0$, denote by $x_{\beta}^{\prime}=(x_{\beta,1},x_{\beta,2},...,x_{\beta,n-1})$ the projection of $x$ to the hyperplane $\{x_{n}=0\}$ along the direction $\beta$, i.e., $x_{\beta,i}=x_{i}-(\beta_{i}/\beta_{n})x_{n}$ for $i=1,2,...,n-1$. Clearly, $|x-(x_{\beta}^{\prime},0)|=(|\beta|/|\beta_{n}|)|x_{n}|$. 6. 6.

B_{r}(x_{0}):=\{x\in R^{n}\big{|}|x-x_{0}|<r\}, $B_{r}:=B_{r}(0)$ and $B_{r}^{+}(x_{0}):=B_{r}(x_{0})\cap R^{n}_{+}$. 7. 7.

T_{r}(x_{0})\ :=\{(x^{\prime},0)\in R^{n}\big{|}|x^{\prime}-x_{0}^{\prime}|<r\}* ** and ** $T_{r}:=T_{r}(0)$.* 8. 8.

$B_{r,h}:=B_{R}(-(R-h)e_{n})$, where $R$ satisfies $(R-h)^{2}+r^{2}=R^{2}$. 9. 9.

$B^{+}_{r,h}$: the spherical cap with the base radius $r$ and height $h$ ($h\leq r$), i.e., $B^{+}_{r,h}:=B_{r,h}\cap R^{n}_{+}$. 10. 10.

$A^{c}:$* the complement of $A$, $\overset{\circ}{A}:$ the interior of $A$ and $\bar{A}:$ the closure of $A$, $\forall A\subset R^{n}$.* 11. 11.

\mathrm{dist}(A,B):=\inf\{|x-y|\big{|}x\in A,y\in B\}, $~{}\forall~{}A,B\subset R^{n}$. 12. 12.

$a^{+}:=\max\{0,a\}$* ** and ** $a^{-}:=-\min\{0,a\}$.* 13. 13.

$\vec{n}(x_{0}):$* the inner normal of $\Omega$ at $x_{0}\in\partial\Omega$.* 14. 14.

$I:$* the unit matrix.* 15. 15.

Given a function $\varphi$. We may use $\varphi_{i}$ or $D_{i}\varphi$ to denote $\partial\varphi/\partial x_{i}$. Similarly, $\varphi_{ij}$ and $D_{ij}\varphi$ denote $\partial^{2}\varphi/\partial x_{i}\partial x_{j}$. 16. 16.

$D\varphi:=(\varphi_{1},...,\varphi_{n})$, $D_{\_}\varphi:=(\varphi_{1},...,\varphi_{n-1})$, $D^{2}\varphi:=\left(\varphi_{ij}\right)_{n\times n}$ and $D^{2}_{\_}\varphi:=\left(\varphi_{ij}\right)_{(n-1)\times(n-1)}$. 17. 17.

For $k\geq 0$, $\varphi$ is called $C^{k,\alpha}$ at $x_{0}$ if there exists a polynomial $P$ of degree $k$ such that $|\varphi(x)-P(x)|\leq K|x-x_{0}|^{k+\alpha}$ for any $x$. Then, $D^{\zeta}\varphi(x_{0}):=D^{\zeta}P(x_{0})$ where $\zeta$ is a multi index, $[\varphi]_{C^{k,\alpha}(x_{0})}:=K$ and $\|\varphi\|_{C^{k,\alpha}(x_{0})}:=K+\sum_{|\zeta|\leq k}|D^{\zeta}\varphi(x_{0})|$. 18. 18.

For viscosity solutions, we use the notations $\bar{S}(\lambda,\Lambda,f)$, $\underline{S}(\lambda,\Lambda,f)$, $S(\lambda,\Lambda,f)$, $M^{+}(M,\lambda,\Lambda)$, $M^{-}(M,\lambda,\Lambda)$ etc. as in C-C.

2 A-B-P maximum principle, Harnack inequality and $C^{\alpha}$ regularity

In this section, we introduce some notations and present some elementary results concerning the viscosity solutions of oblique derivative problems. We say that $u$ touches $v$ by above at $x_{0}$ in $\Omega$ if $u\geq v$ in $\Omega$ and $u(x_{0})=v(x_{0})$. Similarly, we have the definition for touching by below. Now, we give the definition of viscosity solutions.

Definition 2.1

Let $u$ be upper (lower) semicontinuous in $\Omega\cup\Gamma$. We say that $u$ is a viscosity subsolution (supersolution) of1.1 if for any $\varphi\in C^{2}(\Omega\cup\Gamma)$ touching $u$ by above (below) at $x_{0}$ in $\Omega\cup\Gamma$, we have that

[TABLE]

and

[TABLE]

If $u\in C(\Omega\cup\Gamma)$ is both subsolution and supersolution, we call it a viscosity solution.

Remark 2.1

(i) We may write $u\in USC(\Omega\cup\Gamma)$ ($u\in LSC(\Omega\cup\Gamma)$) for short if $u$ is upper (lower) semicontinuous in $\Omega\cup\Gamma$.

(ii) Touching in $\Omega\cup\Gamma$ can be replaced by touching in a neighborhood of $x_{0}$ (see (C-C, Proposition 2.4)). In this case, $\varphi\in C^{2}(\Omega\cup\Gamma)$ in the definition can be replaced by that $\varphi$ is a paraboloid (a polynomial of degree 2).

(iii) The notion of viscosity solutions for oblique derivative problems was introduced first by P.-L. Lions Lio.

Now we present a closedness result concerning the viscosity solutions.

Proposition 2.1

Suppose that $\Gamma\in C^{2}$. Let $\{u_{m}\}$ satisfy

[TABLE]

Suppose that $u_{m}\rightarrow u$ uniformly in any compact subset of $\Omega\cup\Gamma$. Then

[TABLE]

Proof

$u\in S(\lambda,\Lambda,f)$ in $\Omega$ is classical (see Proposition 2.9 C-C). We only prove that $\beta\cdot Du+\gamma u\geq g$ on $\Gamma$ and the proof for the other direction is similar. Let $P$ be a paraboloid touching $u$ by above in a neighborhood of $x_{0}\in\Gamma$. Without loss of generality, we assume that $x_{0}=0$ and $\vec{n}(0)=e_{n}$. Let $\varphi$ denote the representation function of $\Gamma$ near [math] with $\varphi(0)=0$ and $D\varphi(0)=0$. We need to prove that $\beta(0)\cdot DP(0)+\gamma(0)P(0)\geq g(0)$. Suppose not. Then

[TABLE]

By the continuity of $P_{n}$, there exist $\tau,\eta>0$ such that

[TABLE]

By Taylor’s formula, for any $x\in\bar{\Omega}\cap B_{\tau}$,

[TABLE]

By the boundedness of $P_{nn}$, for any $N>0$ there exists $\tau^{\prime}>0$ such that

[TABLE]

The constant $N$ is to be chosen later. Here, we choose $\tau^{\prime}$ small enough such that $\Omega\cap B_{\tau^{\prime}}\subset\subset\Omega\cup\Gamma$ and

[TABLE]

For $\varepsilon>0$, let $\psi_{\varepsilon}(x)=P(x^{\prime},\varphi(x^{\prime}))+(A-\eta/2)(x_{n}-\varphi(x^{\prime}))-N(x_{n}-\varphi(x^{\prime}))^{2}+\varepsilon|x|^{2}$. Then

[TABLE]

Since $u_{m}\rightarrow u$ uniformly in $\Omega\cup\Gamma$, there exists $m_{0}$ large enough such that

[TABLE]

Then

[TABLE]

Hence, $\psi_{\varepsilon}+c_{0}$ touches $u_{m_{0}}$ by above at some $\tilde{x}\in\Omega\cap B_{\tau^{\prime}}$ for a proper $c_{0}$ with $|c_{0}|\leq(\varepsilon\tau^{\prime})^{2}/2.$

By a calculation, the second derivatives of $\psi_{\varepsilon}$ are

[TABLE]

Note that $\varphi(0)=0$ and $D\varphi(0)=0$. Then by choosing $N$ large enough and $\tau^{\prime}$ small enough, we have

[TABLE]

Combining with $M^{+}\left(D^{2}u_{m_{0}}\right)\geq f$, we know that $\tilde{x}\notin{\Omega}$, i.e., $\tilde{x}\in\Gamma$. By the definition of viscosity solutions, we have

[TABLE]

By a calculation (note that $\tilde{x}_{n}=\varphi(\tilde{x}^{\prime})$),

[TABLE]

Hence, at $\tilde{x}$

[TABLE]

Note that $D\varphi(0)=0$ and $|\tilde{x}|\leq\sqrt{\varepsilon}\tau^{\prime}$. Taking $\varepsilon$ small enough, we have

[TABLE]

By the definition of $A$,

[TABLE]

By the continuity of each ingredient in above equation and taking $\varepsilon$ small enough, we have

[TABLE]

which is impossible by recalling2.3.∎

Next, we intend to present an Alexandrov-Bakel’man-Pucci type maximum principle for oblique derivative problems following the idea of M-S.

Lemma 2.1

Let $\Omega\subset B_{1}$ and $u$ satisfy

[TABLE]

Suppose that there exists a direction $\xi\in\partial B_{1}$ such that $\beta\cdot\xi\geq\delta_{1}$ on $\Gamma$. Then

[TABLE]

where $\Gamma_{u}$ is the convex envelop of $u$ in $\Omega$ and $C$ depends only on $n$, $\lambda$, $\Lambda$ and $\delta_{1}$.