Global Regularity for minimal graphs over convex domains in hyperbolic space

TL;DR

This paper establishes that the global regularity of minimal graph solutions over convex domains in hyperbolic space depends solely on convexity, not smoothness, using super-solution construction and invariance properties.

Contribution

It proves the optimal global regularity for minimal graphs over convex domains in hyperbolic space, independent of boundary smoothness, by constructing super-solutions based on invariance.

Findings

Global regularity depends only on convexity

Constructed super-solutions for the problem

Achieved optimal and accurate regularity results

Abstract

In this paper we study the global regularity for the solution to the Dirichlet problem of the equation of minimal graphs over a convex domain in hyperbolic spaces. We find that the global regularity depends only on the convexity of the domain but independent of its smoothness. Basing on the invariance of the problem under translation and rotation transforms, we construct the super-solution to the problem, by which we prove the optimal and accurate global regularity for this problem.

Equations

$$\begin{array}{rl}\Delta u-\frac{u_i{u}_j}{1+|\nabla u{|}^2}{u}_{ij}+\frac{n}{u}& =0\text{in}\Omega ,\\ u& =0\text{on}\partial \Omega ,\\ u& >0\text{in}\Omega ,\end{array}$$

$$[u{]}_{C^{\frac{1}{n+1}}(\overline{\Omega })}\le [(n+1)d_{\Omega }^n{]}^{\frac{1}{n+1}},$$

$$x^{\prime }=(x_1,...,x_{n-1}),|x^{\prime }|=\sqrt{x_1^2+...+x_{n-1}^2}.$$

$$x_0=0,\Omega \subseteq \{(x,x_n)\in R^n|x_n\ge \eta |x^{\prime }{|}^a\}.$$

$$|u{|}_{C^{\bar{a}}(\overline{\Omega })}\le C,$$

$$C=2\sqrt{2R{d}_{\Omega }}\text{when}a=2,C=2(n+1{)}^2{d}_{\Omega }^{\frac{1}{n+1}}\text{when}a=+\infty ,$$

$$x_0=0,\Omega \subseteq \{(x,x_n)\in R^n|x_n\ge \eta |x^{\prime }{|}^a\}.$$

$$|u(x)|\le C|x{|}^{\frac{1}{a+\delta }},\forall x\in \Omega \bigcap \{(x^{\prime },x_n)\in R^n:x^{\prime }=0\}.$$

$$D_{\gamma \gamma }{T}_{\tilde{x}}(\tilde{x})\ge \lambda ,\forall \gamma \in B(0)\subset T_{\tilde{x}}\partial \Omega .$$

$$\Omega =\bigcup _{i=1}^{\infty }{\Omega }_i,$$

$$\Omega \subseteq B_{R+\delta }((R+\delta )e_n).$$

$$\Omega \bigcap P\subseteq B_{R+\delta }((R+\delta )e_n)\bigcap P.$$

$${\gamma }_1=\partial \Omega \bigcap P,{\gamma }_2=\partial B_{R+\delta }((R+\delta )e_n)\bigcap P.$$

$${\gamma }_1(t_0)-{\gamma }_2(t_0)=\underset{t\in [0,\bar{x}]}{\max }({\gamma }_1-{\gamma }_2)(t).$$

$${\gamma }_1^{\prime }(t_0)={\gamma }_2^{\prime }(t_0),0\le {\gamma }_1^{\prime \prime }(t_0)\le {\gamma }_2^{\prime \prime }(t_0),$$

$$\Omega \subseteq B_R(R{e}_n).$$

$$|u(x)|\le M{d_x}^{\alpha },\forall x\in \Omega$$

$$|u{|}_{C^{\alpha }(\overline{\Omega })}\le 2M{d}_{\Omega }^{\alpha }.$$

$$\begin{array}{r}(n-1)\frac{u_r}{r}+\frac{u_{rr}}{1+u_r^2}+\frac{n}{u}=0.\end{array}$$

$$\begin{array}{rl}{U}_r=& \frac{-r}{\sqrt{R^2-r^2}},\\ \frac{U_r}{r}=& \frac{-1}{\sqrt{R^2-r^2}},\\ U_{rr}=& \frac{-R^2}{(R^2-r^2)\sqrt{R^2-r^2}}.\end{array}$$

$$(n-1)\frac{U_r}{r}+\frac{U_{rr}}{1+U_r^2}+\frac{n}{U}=0.$$

$$u(y)\le \sqrt{2R}{d_y}^{\frac{1}{2}},\forall y\in \Omega .$$

$$\begin{array}{rl}u(y)\le U(y)=& \sqrt{R^2-(R-d_y{)}^2}\\ =& \sqrt{2R{d}_y-d_y^2}\\ \le & \sqrt{2R}\sqrt{d_y}.\end{array}$$

$$|u(y)|\le C(a,n,\eta ,d_{\Omega }){d_y}^{\frac{1}{a}},\forall y\in \Omega .$$

$$\Omega \subseteq \{(x,x_n)\in R^n|x_n\ge \eta |x^{\prime }{|}^a\}.$$

$$W(x_1,...,x_n)=((\frac{x_n}{\epsilon }{)}^{\frac{2}{a}}-x_1^2-...-x_{n-1}^2{)}^{\frac{1}{b}},$$

$$W\ge u=0\text{on}\partial \Omega .$$

$$r=|x^{\prime }|=\sqrt{x_1^2+...+x_{n-1}^2},W(x)=W(r,x_n)$$

$$W_i=W_{x_i},W_{ij}=W_{x_i{x}_j}$$

$$\begin{array}{rl}{W}_i=& W_r\frac{x_i}{r},\\ W_{ij}=& W_{rr}\frac{x_i}{r}\frac{x_j}{r}+W_r\frac{{\delta }_{ij}r-\frac{x_i{x}_j}{r}}{r^2},\\ =& \frac{W_r}{r}{\delta }_{ij}+(W_{rr}-\frac{W_r}{r})\frac{x_i}{r}\frac{x_j}{r},\\ W_{in}=& W_{rn}\frac{x_i}{r}.\end{array}$$

Full text

Global Regularity for minimal graphs over convex domains in hyperbolic space

This work was supported by NSFC 11771237.

Huaiyu Jian

Department of Mathematics, Tsinghua University

Beijing 100084, China

( Email: hjian@tsinghua.edu.cn  Tel: 86-10-62772864)

You Li

Department of Mathematics, Tsinghua University

Beijing 100084, China

Abstract: In this paper we study the global regularity for the solution to the Dirichlet problem of the equation of minimal graphs over a convex domain in hyperbolic spaces. We find that the global regularity depends only on the convexity of the domain but independent of its smoothness. Basing on the invariance of the problem under translation and rotation transforms, we construct the super-solution to the problem, by which we prove the optimal and accurate global regularity for this problem.

AMS Mathematics Subject Classification: 35J93, 35B65, 35J25.

Running head: Regularity for Minimal Graphs

Global Regularity for Minimal Graphs over convex domains in Hyperbolic Space

Huaiyu Jian    You Li

1. Introduction

In this paper we consider the Dirichlet problem

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ ($n\geq 2$). In [8], Lin proved that problem (1.1) admits a unique solution $u\in C(\overline{\Omega})\bigcap C^{\infty}(\Omega)$ if $\partial\Omega\in C^{2}$ and $H_{\partial\Omega}\geq 0$, where $H_{\partial\Omega}$ denotes the mean curvature of $\partial\Omega$ with respect to the inward normal direction. Furthermore, he proved that $u\in C^{\frac{1}{2}}(\overline{\Omega})$ if $\partial\Omega\in C^{2}$ and $H_{\partial\Omega}>0$. Anderson [2, 3], Hardt and Lin [6], and Lin [9] studied the various problems related the complete area-minimizing submanifolds in hyperbolic spaces. These results were partially extended by Tonegawa [11] and Guan-Spruck [4] to the constant mean curvature case.

Recently, Han, Shen and Wang in [5] studied the optimal regularity for problem (1.1) and proved the following theorem. See Theorems 1.2 and 3.1 in [5].

Theorem 1.1**.**

Assume that $n\geq 2$, and $\Omega\subseteq R^{n}$ is a bounded convex domain. Then we have

(i)*  (1.1) admits a unique solution $u\in C^{\frac{1}{n+1}}(\overline{\Omega})\bigcap C^{\infty}(\Omega)$, $u$ is concave, and*

[TABLE]

where $d_{\Omega}$ denotes the diameter of $\Omega$;

(ii)*  $u\in C^{\frac{1}{2}}(\overline{\Omega})$ if $\Omega$ is the intersections of finitely many bounded convex $C^{2}$-domains $\Omega_{i}$ with $H_{\partial\Omega_{i}}>0$.*

In this paper we continue to study the global H$\ddot{o}$lder regularity for the concave solutions to problem (1.1), using a method different from [5]. We find that the H$\ddot{o}$lder exponent of the concave solution is independent of the smoothness of $\Omega$ but depends on the convexity of the domain. This fact was found first by us in [7] for a class of singular Monge-Ampere equations. To describe the convexity, we use the concept of $(a,\eta)$- type for bounded convex domain introduced in [7].

From now on, denote

[TABLE]

Definition 1.1[7]. Let $\Omega\subseteq R^{n}$ be a bounded convex domain, $x_{0}\in\partial\Omega$, $a\in[1,+\infty)$ and $\eta>0$. We say $x_{0}$ is $(a,\eta)$- type point if by translation and rotation transforms, we have

[TABLE]

$\Omega$ is called as $(a,\eta)$- type domain if for all $x\in\partial\Omega$, $x$ is $(a,\eta)$- type point; $\Omega$ is called as $(+\infty,\eta)$- type domain if it is not $(a,\eta)$- type for any $a\geq 1$ and any $\eta>0$.

Obviously, a domain which boundary contains pieces of (flat) hyperplane is $(+\infty,\eta)$- type.

Remark 1.1. The convexity requires that the number $a$ should be no less than 1. Since $\Omega$ is bounded and convex, we see that it is sufficient (1.2) holds true only for all $|x^{\prime}|<\mu$ and some small $\mu>0$, and that if $a_{1}>a_{2}$, $(a_{2},\eta_{2})$- type domains are also $(a_{1},\eta_{1})$- type for some $\eta_{1}=C(d_{\Omega},n,a_{1},a_{2},\eta_{2})>0$, as the well-known fact that a $C^{k_{1}}$-domain is also $C^{k_{2}}$-domain if $k_{1}>k_{2}$. Finally, there is no $(a,\eta)$ type domain for $a\in(1,2)$, although the boundary of a convex domain may contains $(a,\eta)$- type points.

The main result of this paper is stated as the following theorem.

Theorem 1.2**.**

Suppose that $\Omega\subseteq R^{n}$ is a bounded convex domain and $u\in C(\overline{\Omega})\bigcap C^{\infty}(\Omega)$ is the concave solution to problem (1.1). If $\Omega$ is $(a,\eta)$- type domain with $a\in[2,+\infty]$, then $u\in C^{\bar{a}}(\overline{\Omega})$ and

[TABLE]

where $\bar{a}=max\{\frac{1}{a},\frac{1}{n+1}\}$ and $C=C(a,\eta,d_{\Omega},n)$ is a positive constant depending only on $a,\eta,n$ and $d_{\Omega}$. In particular,

[TABLE]

where $R$ is the exterior sphere radius of $\Omega$ (see Definition 2.2 below).

We would like to point out that the H$\ddot{o}$lder exponent $\bar{a}$ can not be large than $\frac{1}{n+1}$ if $\Omega$ is $(+\infty,\eta)$- type (see Remark 2.3 in [5] for a domain which boundary contains pieces of (flat) hyperplane ), and it can not be large than $\frac{1}{2}$ if $\Omega$ is $(2,\eta)$- typea (see Lemma 2.5 below for a ball). In this view, our Theorem 1.2 should be optimal and accurate. Unfortunately, the case $a=2$ and $n\geq 3$ of Theorem 1.2 does not include the (ii) of Theorem 1.1, since the domain $\Omega$ described there is not necessarily $(2,\eta)$-type, although the it is $(2,\eta)$-type when $n=2$. Interestingly, the problem (1.1) for $n=2$ also appears in the study of two-dimensional Riemann problems of the Chaplying gas. See section 6 in [10].

For certain points near a $(a,\eta)$ type boundary point for $a\in[1,2)$, we have the following local estimates.

Theorem 1.3**.**

Suppose that $\Omega\subseteq R^{n}$ is a bounded convex domain, $u$ is the concave solution to problem (1.1), and $x_{0}\in\partial\Omega$ is (a, $\eta$) type point with $a\in[1,2)$. Without loss of generality, we assume

[TABLE]

Then for any $\delta>0$, there exists a positive constant $C=C(a,\eta,d_{\Omega},n,\delta)$ such that

[TABLE]

This paper is arranged as follows. In Section 2 we construct a class of non-smooth strictly convex domain which is $(2,\eta)$ type, and prove Theorem 1.2 for the case of $a=2$. In Section 3 we construct super-solutions and prove Theorem 1.2 for the case of $a\in(2,\infty]$. In Section 4, we prove Theorem 1.3. We should point out that applying the invariance of problem (1.1) under translation and rotation transforms to construct super-solutions and using the convexity sufficiently is critical for our arguments, which are different from the proof of Theorem 1.1 in [5].

2. The case $a=2$

For any $\widetilde{x}\in\partial\Omega$, a hyperplane ${\bf P}$ is called a generalized tangent hyperplane (supported hyperplane) at $\widetilde{x}$ if all points of $\Omega$ is on the same side of ${\bf P}$ and $\overline{\Omega}\bigcap{\bf P}\ni\{\widetilde{x}\}$. Obviously, tangent hyperplane at a boundary point is not necessarily unique, unless $\Omega\in C^{1}$. But we can choose a tangent hyperplane ${\bf P}$ such that part of $\partial\Omega$ near the $\widetilde{x}$ is expressed by a function with respect to tangent variable $x^{\prime}\in{\bf P}$. We call such a function as a tangent expression of $\partial\Omega$ near the $\widetilde{x}$, denoted by $T_{\widetilde{x}}\Omega(x^{\prime})$.

If $\Omega\in C^{2}$ and $\widetilde{x}\in\partial\Omega$, the tangent plane at $\widetilde{x}$ is unique which is denoted by $T_{\widetilde{x}}\partial\Omega$, and the tangent expression $T_{\widetilde{x}}\Omega(x^{\prime})$ is locally in $C^{2}$. We see that for a bounded convex $C^{2}$- domain $\Omega$, there is a constant $\lambda$ such that for any $\widetilde{x}\in\partial\Omega$ and any tangent expression $T_{\widetilde{x}}\Omega(x^{\prime})$

[TABLE]

Let $\lambda(\partial\Omega)$ be the maximal number among all such $\lambda$.

We are going to construct a class of non-smooth strictly convex domains which is $(2,\eta)$-type.

Definition 2.1. If $\Omega$ is a bounded domain and there is a $\lambda>0$ and a sequence of $C^{2}$ domains, $\{\Omega_{i}\}_{i=1}^{\infty}$, satisfying $\Omega_{i}\subseteq\Omega_{i+1},\ \lambda(\partial\Omega_{i})\geq\lambda\ \ \forall\,i$, such that

[TABLE]

then $\Omega$ is called as a $\lambda$-convex domain. The maximal number among all such $\lambda$ is denoted by $\lambda(\partial\Omega)$.

Obviously, a $\lambda$-convex domain $\Omega$ may contain singular (angular) points.

Lemma 2.1**.**

Assume $\Omega$ is a $\lambda$-convex domain and $x\in\partial\Omega$. Without loss of generality (by translation or rotation transforms), we assume $x=0$ and $\Omega\subseteq R_{+}^{n}:=\{(x,x_{n})\in R^{n}|x_{n}>0\}$. Then $\Omega\subseteq B_{R}(Re_{n})$, the ball centered in $Re_{n}$ with radius $R$, where $R=\frac{1}{\lambda(\partial\Omega)}$ and $e_{n}=(0,0,\cdots,0,1)$.

Proof.

We will complete the proof by two steps.

Step 1. Assuming that $\Omega$ is smooth, we are going to prove that for any $\delta>0$

[TABLE]

It is enough to prove that for any two-dimensional plane $P$ which contains $x_{n}-$axis, we have

[TABLE]

Note that $\Omega\bigcap P$ and $B_{R+\delta}((R+\delta)e_{n})\bigcap P$ is two dimensional domain in the two-dimensional plane $P$, touched at $x=0$.

Write

[TABLE]

We see that $\gamma_{1}$ and $\gamma_{2}$ is two curves in plane $P$ and we have $curvature(\gamma_{1})\geq\lambda(\partial\Omega)=\frac{1}{R}>\frac{1}{R+\delta}=curvature(\gamma_{2})$. Hence, $\gamma_{1}$ is above (in the inside of) $\gamma_{2}$ near $x=0$ and they are tangent at $x=0$.

To prove (2.2), it is sufficient to prove the claim that $\gamma_{1}$ is always above (in the inner side of) $\gamma_{2}$.

Suppose that the claim is false. We let $\overline{x}\neq 0$ be the nearest contact point from [math] and introduce a new coordinate in plane $P$ as follows. Take line 0$\overline{x}$ to be the coordinate axis with variable t and take the direction orthogonal to the line $0\bar{x}$ to be the graph height coordinate for $\gamma_{1}$ and $\gamma_{2}$. Without loss of generality, we assume $0\geq\gamma_{1}(t)\geq\gamma_{2}(t)$ for $t\in[0,\overline{x}]$.

Now we consider the point $t_{0}$ such that

[TABLE]

Then $t_{0}\in(0,\bar{x})$, $(\gamma_{1}-\gamma_{2})^{\prime}(t_{0})=0$ and $(\gamma_{1}-\gamma_{2})^{\prime\prime}(t_{0})\leq 0$. That is

[TABLE]

which implies that $curvature(\gamma_{1})(t_{0})\leq curvature(\gamma_{2})(t_{0})$, a contradiction!

In this way, we have proved (2.1). Letting $\delta$ to 0, we have

[TABLE]

Step 2. For a general $\lambda$-convex domain $\Omega$, by Definition 2.1 and the result of Step 1, one easily see that (2.3) still holds.

∎

Definition 2.2. We say that a domain $\Omega$ in $R^{n}$ satisfies exterior sphere condition with radius $R$ if for each $x\in\partial\Omega$, there is a ball $B_{R}(y)$ centered at $y$ with radius $R$, such that $B_{R}(y)\supseteq\Omega$ and $\partial B_{R}(y)\bigcap\partial\Omega\ni x$.

It is direct from Lemma 2.1 that

Lemma 2.2**.**

A $\lambda$-convex domain satisfies exterior sphere condition with radius $R=\frac{1}{\lambda(\partial\Omega)}$.

Lemma 2.3**.**

A bounded convex domain satisfying exterior sphere condition with radius $R$ is $(2,\frac{1}{2R})$- type domain. Conversely, a $(2,\eta)$- type domain satisfies exterior sphere condition with radius $R=max\{\frac{1}{\eta},\ d_{\Omega}\}$.

Proof.

See Lemma 2.1 in [7]. ∎

Therefore, $\lambda$-convex domains are $(2,\eta)$- type.

Lemma 2.4**.**

Let $\Omega$ be a bounded convex domain and $u$ be a concave function with $u|_{\partial\Omega}=0$. If there are $\alpha\in(0,1]$ and $M>0$ such that

[TABLE]

where $d_{x}=dist(x,\partial\Omega)$, then $u\in C^{\alpha}(\overline{\Omega})$ and

[TABLE]

Proof.

See Lemma 2.3 in [7]. ∎

Lemma 2.5**.**

When $\Omega=B_{R}(0)\subset R^{n}$, then $U(x)=\sqrt{R^{2}-|x|^{2}}$ is the solution to problem (1.1).

Proof.

In the spherical symmetry case, $u(x)=u(r)$ where $r=|x|$. Then the equation in (1.1) is reduced to

[TABLE]

By a simple computation, we obtain

[TABLE]

Using (2.6), we have

[TABLE]

∎

Theorem 2.6**.**

Suppose $\Omega$ is a bounded convex domain satisfying exterior sphere condition with radius $R$ and $u$ is the concave solution to problem (1.1). Then $u\in C^{\frac{1}{2}}(\overline{\Omega})$, and $|u|_{C^{\frac{1}{2}}(\overline{\Omega})}\leq 2\sqrt{2Rd_{\Omega}}$.

Proof.

Due to Lemma 2.4, it is enough to prove

[TABLE]

Taking a $y\in\Omega$, we can find $z\in\partial\Omega$ such that $dist(y,z)=d_{y}.$ Without loss generality (by translation and rotation), we assume $z=0$ and the line determined by $z$ and $y$ is the $x_{n}-axis$. Note that the equation in problem (1.1) is invariant under translation and rotation transforms.

Since $\Omega$ is a bounded convex domain satisfying exterior sphere condition with radius $R$, we conclude that $\Omega\subseteq B_{R}(Re_{n})$. Let $U$ be the solution of (1.1) in $B_{R}(Re_{n})$ as in Lemma 2.5. By comparison principal, we conclude $u\leq U$ in $\Omega$. Restricting on the point $y$ we have

[TABLE]

∎

Proof of the case $a=2$ of Theorem 1.2:   it is direct from Lemma 2.3 and Theorem 2.6.

3. Proof of Theorem 1.2

In previous Section we have proved Theorem 1.2 for the case $a=2$. In this section, we prove Theorem 1.2 for the case $a\in(2,\infty]$ and thus complete its proof.

Theorem 3.1**.**

Let $\Omega$ be $(a,\eta)$ type domain with $a\in(2,+\infty)$ and $u$ is the concave solution to problem (1.1). Then $u\in C^{\frac{1}{a}}(\overline{\Omega})$, and $|u|_{C^{\frac{1}{a}}(\overline{\Omega})}\leq C(a,\eta,d_{\Omega},n)$.

Proof.

By Lemma 2.4, it is sufficient to prove

[TABLE]

For any $y\in\Omega$, we can find $z\in\partial\Omega$, such that $dist(y,z)=d_{y}.$ Since the domain $\Omega$ is $(a,\eta)$- type, without loss generality (by translation and rotation), we may assume $z=0$, and take the line determined by $z$ and $y$ as the $x_{n}-axis$ such that

[TABLE]

We should point out that problem (1.1) is invariant under translation and rotation transforms.

Let

[TABLE]

where $b\geq 2$ is a constant to be fixed. We will choose a $\varepsilon>0$ such that $W$ is a super-solution to problem (1.1).

Observing that (3.2) implies that $\partial\Omega$ lies over the hypersurface $x_{n}=\eta|x^{\prime}|^{a}$, we can find a small $\varepsilon=C(a,\eta,d_{\Omega},n)>0$ such that

[TABLE]

For brevity, we write

[TABLE]

and

[TABLE]

for $i,j\in\{1,2,...,n-1\}$. A direct computation yields

[TABLE]

Let $F[u]$ denote the left hand side of the equation in (1.1), i.e.,

[TABLE]

Then by (3.5) we have

[TABLE]

where

[TABLE]

By the expression (3.3), we compute

[TABLE]

Hence

[TABLE]

[TABLE]

and

[TABLE]

Adding the above three equalities, we obtain

[TABLE]

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Therefore, we obtain

[TABLE]

Observe the pairs of $J_{1}$ and $J_{7}$, $J_{2}$ and $J_{8}$ , $J_{3}$ and $J_{4}$. Each pair can be combined or canceled if we take $b=2$. Now we fix $b=2$. Consequently, we have

[TABLE]

Hence,

[TABLE]

Recalling that

[TABLE]

we have

[TABLE]

which implies that

[TABLE]

and

[TABLE]

Therefore, we obtain

[TABLE]

Since $a\in(2,+\infty)$, $2-a<0$ and $\frac{2}{a}-2<-1<0$, we have

[TABLE]

Finally, choosing a smaller $\varepsilon=C(a,d_{\Omega})>0$ (if necessary) such that

[TABLE]

we have

[TABLE]

By this and (3.4), we have proved that $W$ is an super-solution to problem (1.1). By comparison principal we have

[TABLE]

In particular, for all $(\mathbf{0},x_{n})\in\Omega$ we have

[TABLE]

Note that $d_{y}=y_{n}$ and $y=(0,y_{n})$ by the choice of the coordinate in the beginning. It follows from (3.9) that

[TABLE]

which proves (3.1) and thus completes the proof of Theorem 3.1. ∎

Theorem 3.2**.**

If $\Omega$ is a general bounded convex domain and $u$ is the concave solution to problem (1.1), then $u\in C^{\frac{1}{n+1}}(\overline{\Omega})$ and

[TABLE]

Proof.

This is the result (i) of Theorem 1.1 except for a different Hölder norm, but our argument provides another proof.

To prove (3.10), as the arguments between (3.1) and (3.2), we take any $y\in\Omega$ and choose $z\in\partial\Omega$ such that $dist(y,z)=d_{y}.$ Since the domain $\Omega$ is convex, we may assume $z=0$, and take the line determined by $z$ and $y$ as the $x_{n}-axis$ such that $\Omega\subseteq R_{+}^{n}$. Without loss of generality, we assume $d_{\Omega}\leq 1$. Otherwise, we can use the transform

[TABLE]

to arrive at the assumed case. Let

[TABLE]

Then

[TABLE]

By direct computation we have

[TABLE]

Since $d_{\Omega}\leq 1$ and $x_{n}\in[0,1]$, we have

[TABLE]

and

[TABLE]

Hence,

[TABLE]

which implies $U$ is a supper-solution to problem (1.1). Therefore

[TABLE]

which, together with Lemma 2.4, implies (3.10). ∎

Proof of (i) of Theorem 1.2:   When $a\in(2,n+1]$, Theorem 3.1 is Theorem 1.2 exactly. When $a\in(n+1,+\infty]$, Theorem 1.2 follows from Theorem 3.2.

4. Proof of Theorem 1.3

In this section, we assume that $u$ is a concave solution to problem (1.1) and $x_{0}\in\partial\Omega$ is $(a,\eta)$- type point with $a\in[1,2)$ satisfying (1.5). We are going to prove Theorem 1.3.

Since $(1,\eta)$- type point is of course $(1+\varepsilon,\eta(\varepsilon))$- type point for any $\varepsilon>0$, it is sufficient to prove (1.6) for any $a\in(1,2)$. Hence, from now on we assume $a\in(1,2)$.

We will choose a small positive number $A$ such that

[TABLE]

Let

[TABLE]

We use $\widetilde{\Omega}$ to denote the image of $\Omega$ under this transform. It is easy to check $\widetilde{u}$ is the solution to problem (1.1) in the domain $\widetilde{\Omega}$ which satisfies

[TABLE]

Since $x_{0}=0\in\partial\Omega$, by the convexity and (4.2) we see that $\widetilde{\Omega}\subseteq\Omega$. Note that the surface $x_{n}=\eta|x^{\prime}|^{a}$ is transformed to the surface

[TABLE]

Then by (4.1)we have

[TABLE]

For brevity, we denote $\widetilde{u}$ and $\widetilde{\Omega}$ still by $u$ and $\Omega$, respectively, in the following. Let

[TABLE]

where $b\in(2,3)$ can be arbitrary constant. We will prove $W$ is a super-solution to problem (1.1) in $\Omega$.

Taking $\varepsilon=1$ in (3.3) we obtain the function as in (4.4). Hence have (3.7)-(3.8) where $\varepsilon=1$. In this case, by (4.4) we have

[TABLE]

Recalling that $a\in(1,2)$, $b\in(2,3)$, $|x^{\prime}|^{a}\leq x_{n}\leq A$ in $\Omega$ by (4.3) and $W^{b}(x)=x_{n}^{\frac{2}{a}}-|x^{\prime}|^{2}$ by (4.4), we obtain

[TABLE]

[TABLE]

Hence, in $\Omega$ we have

[TABLE]

Since

[TABLE]

by (4.3) and (4.4), we obtain

[TABLE]

Furthermore, it follows from the fact $n\geq 2,a\in(1,2)$ and $b\in(2,3)$ that

[TABLE]

Therefore, we can take $A>0$ is small enough in advance such that $\Phi\leq 0$. Hence

[TABLE]

In this way, we have proved that $W$ is a super-solution to problem (1.1) in $\Omega.$ By comparison principal, $0\leq u\leq W$. Restricting this on to $x_{n}-axis$, we obtain that

[TABLE]

for any $b\in(2,3)$, which implies the desired (1.6) for any $\delta\in(0,\frac{1}{2})$. Moreover, this implies (1.6) for any $\delta\geq\frac{1}{2}$. The proof of Theorem 1.3 is finished.