Refined Asymptotics for Minimal Graphs in the Hyperbolic Space

TL;DR

This paper investigates the boundary behavior of solutions to the minimal graph Dirichlet problem in hyperbolic space, especially near singular boundary points, providing refined asymptotic estimates for the case when n=2.

Contribution

It characterizes boundary behaviors at singular points and offers refined asymptotic estimates for solutions in hyperbolic space, advancing understanding of minimal graphs with complex boundary conditions.

Findings

Boundary behavior characterized at singular points

Refined estimates obtained for n=2 case

Enhanced understanding of minimal graphs in hyperbolic space

Abstract

We study the boundary behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries and characterize the boundary behaviors of $f$ at the points strictly located in the tangent cones at the singular points on the boundary. For $n=2$, we also obtain a refined estimate of $f.$

Equations

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

$$\bigg{|}\left(\frac{H_{\partial\Omega}}{2d}\right)^{\frac{1}{2}}f-1\bigg{|}\leq Cd^{\frac{\alpha}{2}},$$

$$|f(x)-f_V(x)|\le Cf(x)|x-x_0|,$$

$${\Omega }_{x_0,\mu ,{\kappa }_1,{\kappa }_2}=B_{\frac{1}{{\kappa }_1}}\left(x_0+\frac{1}{{\kappa }_1}{\nu }_1\right)\bigcap B_{\frac{1}{{\kappa }_2}}\left(x_0+\frac{1}{{\kappa }_2}{\nu }_2\right),$$

$$\left|f(x)-f_{\ast }(x)\right|\le Cf(x)|x-x_0{|}^{1+\alpha -ϵ},$$

$${\overline{V}}_{\mu }$$

$$f=rh(\theta ),$$

$$\frac{h^{\prime \prime }+h}{r}-\frac{h^{\prime 2}(h^{\prime \prime }+h)}{r(1+h^2+h^{\prime 2})}+\frac{n}{rh}$$

$$\mathcal{L}h$$

$$\mathcal{L}\left(A(\sin \frac{\theta }{\mu }{)}^{\frac{1}{1+\alpha }}+B(\sin \frac{\theta }{\mu }{)}^{\frac{1}{1+\beta }}\right)$$

$$\phi (\theta )={\left(\sin \frac{\theta }{\mu }\right)}^{\frac{1}{1+\alpha }}.$$

$${\phi }^{\prime }=\frac{{\phi }^{-\alpha }}{1+\alpha }\frac{1}{\mu }\cos \frac{\theta }{\mu },{\phi }^{\prime \prime }=-\frac{1}{{\mu }^2(1+\alpha {)}^2}\phi -\frac{\alpha }{{\mu }^2(1+\alpha {)}^2}{\phi }^{-1-2\alpha }.$$

$$\mathcal{L}(A\phi )$$

$$+n\left[1+A^2{\phi }^2+\frac{A^2}{{\mu }^2(1+\alpha {)}^2}{\phi }^{-2\alpha }(1-{\phi }^{2+2\alpha })\right].$$

$$\mathcal{L}(A\phi )$$

$$+A^4{\phi }^2\left[(1-\frac{1}{(1+n{)}^2{\mu }^2}){\phi }^2-\frac{n}{(1+n{)}^2{\mu }^2}{\phi }^{-2n}\right].$$

$$\mathcal{L}(\sqrt{(1+n)\mu }\phi )\le n-n{\phi }^{2-2n}\le 0.$$

$$\psi$$

$$h$$

$$\mathcal{L}(A(\sin \frac{\theta }{\mu }{)}^{\frac{1}{1+\alpha }}+B(\sin \frac{\theta }{\mu }{)}^{\frac{1}{1+\beta }})\le A^2\frac{n-\alpha }{{\mu }^2(1+\alpha {)}^2}{\phi }^{-2\alpha }+n+C{A}^2{\phi }^{-2\alpha +\tau },$$

$$T_L(x_1,\cdots ,x_{n+1})=\frac{L}{(x_1-L{)}^2+x_2^2+...+x_{n+1}^2}(L^2-|x{|}^2,2L{x}_2,\cdots ,2L{x}_{n+1}).$$

$$\frac{2L^2{x}_{n+1}}{(x_1-L{)}^2+\cdot \cdot \cdot +x_{n+1}^2}\to 0\text{as }{x}_1^2+\cdot \cdot \cdot +x_{n+1}^2\to \infty .$$

$$f_V(x)=|x|g_V(\theta ),$$

$$\Omega \cap B_R(x_0)=\{x\in B_R:x_n>f(x^{\prime })\},$$

$$V_{{\theta }_0}\subseteq \overline{\Omega },-V_{{\theta }_0}\subseteq {\Omega }^c.$$

$$B_i^L=B_{\frac{L/2}{<{\nu }_i,e_n>}}\left(x_0+\frac{L/2}{<{\nu }_i,e_n>}{\nu }_i\right).$$

$$\bigcap _i\partial B_i^L=\{x_0,q\},$$

$$\frac{L}{2}\le \frac{L/2}{<{\nu }_i,e_n>}\le \frac{L/2}{\sin {\theta }_0}.$$

$$\tilde{\Omega }=\bigcap _i{B}_i^{2L}\subset \Omega ,$$

$$\text{dist}(x,\partial V)\ge \delta |x-x_0|,$$

$$\text{dist}(x,\partial \tilde{\Omega })>\frac{\delta }{2}|x-x_0|.$$

$$x_0=(-L,0,\cdot \cdot \cdot ,0),q=(L,0,\cdot \cdot \cdot ,0).$$

Full text

Authors acknowledge the support of NSFC Grant 11571019.

1. Introduction

Assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded domain. Lin [10] studied the Dirichlet problem of the form

[TABLE]

Geometrically, the graph of $f$ is a minimal surface in $\mathbb{H}^{n+1}$ with its asymptotic boundary at infinity given by $\partial\Omega$. For $n=2$, (1.1) also appears in the study of the Chaplygin gas. See [12] for details.

The existence of a unique solution $f\in C(\bar{\Omega})\bigcap C^{\infty}(\Omega)$ to (1.1) was shown in [10] with the assumption that $\Omega\subset\mathbb{R}^{n}$ is a $C^{2}$-domain and its boundary has nonnegative mean curvature $H_{\partial\Omega}$ with respect to the inward normal direction of $\partial\Omega$. Concerning the higher global regularity, Lin proved if $H_{\partial\Omega}>0$, then $f\in C^{1/2}(\bar{\Omega})$. In [6], Han and we proved that under the condition $H_{\partial\Omega}\geq 0$, $f\in C^{\frac{1}{n+1}}(\bar{\Omega}).$ Han and we also proved in [6] that (1.1) admits a unique solution $f\in C^{1/2}(\bar{\Omega})\bigcap C^{\infty}(\Omega)$ under the assumption that $\Omega$ is the intersection of finitely many bounded convex $C^{2}$-domains $\Omega_{i}$ with $H_{\partial\Omega_{i}}>0$.

Concerning asymptotic behaviors of solutions to (1.1), when $\Omega$ is sufficiently smooth, the expansion near the boundary of solution to the Dirichlet problem for minimal graphs in the hyperbolic space is shown in [3]. When $\Omega$ is singular, Han and we [7] studied the asymptotic behaviors of solution $f$ on $\Omega\subset\mathbb{R}^{2}$ whose boundary are piecewise regular with positive curvatures and derived an estimate of $f$ by comparing it with the corresponding solutions in the intersections of interior tangent balls.

The boundary geometry has great effects on behaviors of solutions to (1.1). When the boundary is regular, asymptotic behaviors are much clearer. For example, if $\Omega$ is a bounded $C^{2,\alpha}$-domain with $H_{\partial\Omega}>0$, for some $\alpha\in(0,1)$, then

[TABLE]

where $d$ is the distance function to $\partial\Omega$. Another problem involving positive boundary curvatures is discussed by Jian and Wang [9]. However, difficulties arise when we study asymptotic behaviors of solutions $f$ in domains with singularity. In the general case of singular boundary, it is natural to compare solutions with the corresponding solutions in the tangent cones. This is the approach Han and the first author adopted in the study of the Liouville equation in [4] and the Loewner-Nirenberg problem in [5]. However, for (1.1) in domains with singularity, in light of (1.2), we should abandon this approach, since the boundaries of tangent cones bounded by finitely many hyperplanes have zero mean curvature wherever they are smooth. In a sense, we need to preserve the positivity of the boundary mean curvature. For $n=2$, in domains whose boundaries are piecewise regular with positive curvatures, Han and we [7] studied the asymptotic behaviors of $f$ to (1.1) and proved that $f$ can be well approximated by the corresponding solutions in the intersections of interior tangent balls.

In this paper, we continue our study of the boundary behaviors of solutions $f$ to (1.1) in general convex domains with singular asymptotic boundaries. We characterize the boundary behaviors of $f$ at the points strictly located in the tangent cones at the singular points on the boundary and prove that $f$ at these points can be well approximated by the corresponding solutions in tangent cones. For $n=2$, we also obtain a refined estimate of $f$. From the results in this paper and also the results in [6] and [7], we can see the degeneracy of (1.1) has great effect on the boundary behaviors of the solution, which we can compare with the results in [11].

Our first main theorem in this paper is the following result.

Theorem 1.1**.**

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^{n}$ and, for some $x_{0}\in\partial\Omega$ and $R>0$, $\partial\Omega\cap B_{R}(x_{0})$ consist of $k$ $C^{1,1}$-hypersurfaces $S_{i},\,i=1,\cdots,k,$ with the angle between any two of the tangent planes at $x_{0}$ less than $\pi$. Suppose $f\in C(\bar{\Omega})\cap C^{\infty}(\Omega)$ is the solution of (1.1) in $\Omega$ and $f_{V}$ is the corresponding solution in the tangent cone $V$ of $\Omega$ at $x_{0}$. Then, for any $\delta>0$ and any $x\in\Omega$ close to $x_{0}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|$,

[TABLE]

where $C$ is some constant depending only on $\delta$ and the geometry of $\partial\Omega$ near $x_{0}.$

Inspired by results in [7], we now compare solutions $f$ to (1.1) with those in the intersections of interior tangent balls and prove a refined estimate.

Theorem 1.2**.**

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^{2}$ and, for some $x_{0}\in\partial\Omega$ and $R>0$, $\partial\Omega\cap B_{R}(x_{0})$ consist of two $C^{2,\alpha}$-curves $\sigma_{1}$ and $\sigma_{2}$ intersecting at $x_{0}$ with an angle $\mu\pi$, for some constants $\alpha\in(0,1)$ and $\mu\in(0,1)$. Assume the curvature $\kappa_{i}$ of $\sigma_{i}$ at $x_{0}$ is positive, $i=1,2.$ Suppose $f\in C(\bar{\Omega})\cap C^{\infty}(\Omega)$ is the solution of (1.1) in $\Omega$ and $f_{*}$ is the corresponding solution in

[TABLE]

where $\nu_{1}$ and $\nu_{2}$ are interior unit normal vector to $\sigma_{1}$ and $\sigma_{2}$ at $x_{0}$, respectively. Then, for any $\epsilon\in(0,\alpha)$ and $\delta>0$, there exists a constant $\mu({\epsilon,\alpha})>0$, such that, if $\mu\leq\mu({\epsilon,\alpha})$, then, for any $x\in\Omega$ close to $x_{0}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|$,

[TABLE]

where $C$ is a positive constant depending only on $R$, $\mu$, $\alpha$, $\epsilon$, $\delta$ and the $C^{2,\alpha}$-norms of $\sigma_{1}$ and $\sigma_{2}$ in $B_{R}(x_{0})$.

The paper is organized as following. In Section 2, we study the boundary behaviors of solutions of (1.1) in bounded convex domains bounded by finitely many $C^{1,1}$-hypersurfaces and prove Theorem 1.1. In Section 3, we study $f$ in domains whose boundaries are piecewise regular with positive curvatures and prove Theorem 1.2.

2. Solutions in Convex Domains Bounded by Hypersurfaces

In this section, we discuss the boundary behaviors of solutions of (1.1) in convex domains bounded by several $C^{1,1}$ hypersurfaces. We prove, at points strictly located in tangent cones defined at singular points on the boundary, the solutions $f$ are well approximated by the corresponding solutions in these cones.

First, we discuss (1.1) in infinite cones and prove the existence and uniqueness of solutions of (1.1) in infinite cones. Since this part follows [7] closely, we only sketch the proof.

For some constant $\mu\in(0,1)$, define

[TABLE]

This is an infinite cone in $\mathbb{R}^{2}$, expressed in polar coordinates. Then, $V_{\mu}:=\overline{V}_{\mu}\times\mathbb{R}^{n-2}$ is an infinite cone in $\mathbb{R}^{n}$. Our goal is to find a solution $f$ to (1.1) in $\Omega=V_{\mu}$, whose form is given by

[TABLE]

where $(r,\theta)$ is the polar coordinates in $\mathbb{R}^{2}.$ Substituting (2.2) in $\eqref{eq-MinGmain}$, we have

[TABLE]

In view of (2.3), we set the operator $\mathcal{L}$ acting on functions $h=h(\theta)$, $\theta\in(0,\mu\pi)$, by

[TABLE]

First, we construct supersolutions of $\mathcal{L}$.

Lemma 2.1**.**

For some constant $\mu\in(0,1)$, there exist constants $A>0$, $B\geq 0$, $\alpha\in[n,+\infty)$ and $\beta\in(0,1)$ such that

[TABLE]

Proof.

For some $\alpha>0$, set

[TABLE]

By differentiating twice, we have

[TABLE]

Then, for some positive constant $A$,

[TABLE]

We first consider the case $\mu\leq\frac{1}{1+n}$. With $\alpha=n$, we have

[TABLE]

Hence,

[TABLE]

Next, we consider the case $\mu>\frac{1}{1+n}$. Fix an arbitrary constant $\alpha\in(n,+\infty).$ Set

[TABLE]

where we take $\beta=\min\{\frac{1}{2}(\frac{1}{\mu}-1),\frac{1}{100}\},$ $A\geq 1$ to be determined, and set $B=CA$ for a sufficiently large constant $C$ to be determined. We can compare with the corresponding terms appearing in the proof of Lemma 2.1 in [7]. Then, we proceed similarly as in the proof of Lemma 2.1 in [7] and we just point out a key difference that, for some positive constant $\tau$, when $\sin\frac{\theta}{\mu}<\frac{1}{1+\alpha},$

[TABLE]

and $n-\alpha<0.$ Hence we obtain the desired result. ∎

For any $L>0,$ we define an operator $T_{L}$ by

[TABLE]

Then $T_{L}$ is an isometric automorphism in $\mathds{H}^{n+1},$ which maps $(L,0,\cdots,0)$ to infinity. Restricted to $\mathbb{R}^{n}\times{\{x_{n+1}=0\}},$ $T_{L}$ is a conformal transform. We can obtain (2.7) by a combination of some conformal transforms in $\mathbb{R}^{n+1}.$ (See [7]). It is obvious that

[TABLE]

With Lemma 2.1 and $T_{L},$ we prove the existence and uniqueness of the solution of (1.1) in any cone $V\subseteq\mathbb{R}^{n}$ by following closely the proof of Theorem 2.3 in [7]. In fact, any cone $V$ is contained in a cone $\overline{V}$ bounded by two hyperplanes with a angle less than $\pi$ and the super-solution to (1.1) on $\overline{V}$ is a upper bound for the solution to (1.1) on $V$ by the maximum principle. From the proof, we also conclude that the solution in $V$ has the form

[TABLE]

with $\theta\in\mathds{S}^{n-1}$.

Next, we turn our attention to (1.1) on domains.

Let $\Omega$ be a bounded convex domain and, for some $x_{0}\in\partial\Omega$ and $R>0$, $\partial\Omega\cap B_{R}(x_{0})$ consist of $k$ $C^{1,1}$-hypersurfaces $S_{i}$, $i=1,\cdots,k$, with the angle between any two of the tangent planes at $x_{0}$ less than $\pi$. Denote by $V_{x_{0}}$ the tangent cone of $\Omega$ at $x_{0}.$ Then, $V_{x_{0}}$ is bounded by $P_{i}$, the tangent plane of $S_{i}$ at $x_{0},$ for $i=1,\cdots,k.$ Denote by $\nu_{i}$ the unit inner normal vector to $P_{i}$, $i=1,\cdots,k$.

Assume $x_{0}\in\partial\Omega$ is the origin 0. By the convexity, $\Omega$ is a bounded Lipschitz domain and we can assume

[TABLE]

for some Lipschitz function $f$ on $B^{\prime}_{R}\subset\mathbb{R}^{n-1}$, with $f(0)=0$. Then, there exists a finite circular cone $V_{\theta_{0}}$ such that $x_{0}$ is its vertex, the $x_{n}$-axis its axis, $2\theta_{0}$ the apex angle, $h$ the height, and

[TABLE]

In the following, we denote by $\mu_{x_{0}}\pi$ the minimal angle among angles between any two of the tangent planes at $x_{0}$.

For a positive constant $L$, set

[TABLE]

It is easy to see that

[TABLE]

where $q=x_{0}+Le_{n}$. Note $<\nu_{i},e_{n}>\,\,\geq\sin\theta_{0}.$ Hence,

[TABLE]

For some constant $L$ depending only on $R$ and the $C^{1,1}$-norms of $S_{i}$, for $i=1,\cdots,k,$ we note that each ball $B_{i}^{L}$ is above the corresponding hypersurface $S_{i}$, although it is not necessarily in $\Omega$.

Now we are ready to prove Theorem 1.1

Proof of Theorem 1.1.

Throughout the proof, we always denote by $C$ some positive constant depending only on $n$, $R$, $\theta_{0}$, $\delta$, $\mu_{x_{0}}$, $h$ and the $C^{1,1}$-norms of hypersurfaces $S_{i},\,i=1,\cdots,k$, near $x_{0}$. Set, for $L$ sufficiently small,

[TABLE]

where $B_{i}^{2L}$ is defined above.

Then, for $|x-x_{0}|$ small with $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|$, we have

[TABLE]

and

[TABLE]

For convenience, we rotate the coordinates such that $x_{n}$-axis above becomes $x_{1}$-axis and assume

[TABLE]

Let $\widetilde{f}$ be the solution of (1.1) in $\widetilde{\Omega}$. The maximum principle implies

[TABLE]

We note that the tangent cone of $\Omega$ at $x_{0}$ is also the tangent cone of $\widetilde{\Omega}$ at $x_{0}$. We consider the map $T_{L}$ introduced in (2.7). Then, $T_{L}|_{\mathbb{R}^{n}\times\{0\}}$ maps conformally $\widetilde{\Omega}$ to an infinite cone $\widetilde{V}$, which conjugates to $V$, with

[TABLE]

and $T_{L}$ maps the minimal graph $\{(x,\widetilde{f}(x))\}$ in $\mathds{H}^{n+1}$ to the minimal graph $\{(y,\widetilde{f}_{\widetilde{V}}(y))\}$ in $\mathds{H}^{n+1}$. By (2.7) and (2.8), we have

[TABLE]

and, for $|x-x_{0}|$ small and $a\in\{2,\cdot\cdot\cdot,n\},$

[TABLE]

and

[TABLE]

By (2.8), when $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|,$

[TABLE]

and

[TABLE]

where we used the fact that $g_{\widetilde{V}}(\theta)\geq c$, for some positive constant $c$ depending on ${\widetilde{V}}$ and $\delta$, when $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|$ and $x$ is close to $x_{0}$, by noting $g_{\widetilde{V}}(\theta)>0.$ Therefore, combining (2.11) and the fact $g_{\widetilde{V}}=g_{V}$ by (2.12), we have

[TABLE]

Also, by the maximum principle, we have, for any $x\in\Omega,$

[TABLE]

This finishes the proof. ∎

3. Refined expansion

In [7], we studied asymptotic behaviors of $f$ in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures and approximated such solutions by the corresponding solutions in the intersections of interior tangent balls up to an order $|x|^{\beta}$, with $\beta\in(0,\frac{\alpha}{2}]$. On the other hand, Theorem 1.1 demonstrates that, at points strictly located in tangent cones defined at the singular points on the boundary, the solutions $f$ are well approximated by the corresponding solutions in these cones up to the order $|x|$. In light of this, we expect that the corresponding solutions in the interior tangent balls should provide a refined estimate over the estimate in [7].

To this end, we need a localization lemma which provides more information on the local properties of asymptotic expansions near singular boundary points up to certain orders. Compare with Lemma 3.1 in [7].

Lemma 3.1**.**

Let $\Omega$ and $\Omega_{*}$ be two convex domains in $\mathbb{R}^{2}$ such that, for some $x_{0}\in\partial\Omega$ and some $R_{0}\in(0,1]$,

[TABLE]

and that $\partial\Omega\cap B_{R_{0}}(x_{0})$ consists of two $C^{1,1}$-curves $\sigma_{1}$, $\sigma_{2}$ intersecting at $x_{0}$, with the angle between the tangent lines of $\sigma_{1}$ and $\sigma_{2}$ given by $\mu\pi,$ for some $\mu\in(0,1).$ Suppose that $f$ and $f_{*}$ are solutions of (1.1) for $\Omega$ and $\Omega_{*}$, respectively. Then, for any $\beta>0$ and $\delta>0$, there exists a constant $\mu(\beta)$ such that, for any $\mu\in(0,\mu(\beta)]$ and any $x\in\Omega$ close to $x_{0}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|$,

[TABLE]

where $C$ is a positive constant depending only on $\mu$, $\delta$ and the $C^{1,1}$-norms of $\sigma_{1}$ and $\sigma_{2}$ in $B_{R_{0}}(x_{0})$.

Proof.

We note that the equation in (1.1) is invariant under the scaling $f\mapsto f(R\cdot)/R$. Without loss of generality, we assume $x_{0}=0$ and $R_{0}=1$ and prove, for any $x\in\Omega$ close to $x_{0}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x|$,

[TABLE]

For any $x\in\Omega\cap B_{r_{0}}$ with $\text{dist}(x,\partial\Omega)\geq\delta|x|,$ we have

[TABLE]

where $r_{0}$ and $c$ are small positive constants obtained by Theorem 1.1 and $C_{\mu}$ is a positive constant obtained by the maximum principle and (2.8). Hence, for any $r_{1}\in(0,r_{0})$, (3.2) holds for any $x\in\Omega\cap(B_{r_{0}}\setminus B_{r_{1}})$, with $\text{dist}(x,\partial\Omega)\geq\delta|x|$ by taking $C$ in (3.2) large.

Let $g$ be the solution to (1.1) in $\Omega\cap B_{1}$. By the maximum principle, we have

[TABLE]

Write $r=|x|$. We claim, there exists a small $r_{1\mu}$ such that

[TABLE]

By combining (3.3), (3.4), and (3.5), we have, for any $x\in\Omega\cap B_{r_{1\mu}}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x|$,

[TABLE]

Hence, we obtain (3.2) for any $x\in\Omega\cap B_{r_{1\mu}}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x|$.

We now prove the first inequality in (3.5). First, we consider the boundary condition. Proceeding as in [7], we have

[TABLE]

and

[TABLE]

where $r_{0\mu}$ is the small positive constant defined in Lemma 3.1 in [7] and the subscript $\mu$ indicates its dependence on $\mu.$ In the following, we assume $\mu$ is small. Then by (3.7),

[TABLE]

Next, we require, for some small $r_{1\mu}<r_{0\mu},$

[TABLE]

To this end, take

[TABLE]

Combining with (3.6), the boundary condition is satisfied.

Next, set

[TABLE]

and

[TABLE]

We will prove $Q(h)\geq 0$ in $\Omega\cap B_{r_{1\mu}}$, for the general dimension $n$. Take $r_{1\mu}$ sufficiently small, with ${r_{1\mu}}\ll{r_{0\mu}}$. We have, for $r\leq r_{1\mu},$

[TABLE]

We claim that

[TABLE]

where $C$ is a positive constant independent of $\mu.$ In fact, we have $C=2.1+1.2(n-1)$ from the proof of (3.11). Assuming (3.11), we have, by (3.7),

[TABLE]

By (3.7), we choose $\mu$ small so that $C_{\mu}$ is small. Therefore, $Q(h)\geq 0.$

Now we prove (3.11). Note that

[TABLE]

are invariant under constant orthogonal transforms. Hence, in a neighborhood of any point $p\in\Omega\cap B_{r_{1_{\mu}}},$ by a rotation, we can assume $\nabla h(p)=h_{1}(p)$ and proceed to calculate at $p$ in such coordinates. Set $\imath,\jmath\in\{2,\cdot\cdot\cdot,n\}$ and

[TABLE]

Then,

[TABLE]

where we used the fact that $h_{\imath}=0$ implies

[TABLE]

Note $a_{ii}(h),a_{ii}(f)$ are nonnegative by definition. Hence, by (3.15) again and (3.10),

[TABLE]

and hence

[TABLE]

and

[TABLE]

Next, we consider $a_{ij}(f)f_{ij}$ for $i\neq j.$ Note that $i\neq j$ implies $i\neq 1$ or $j\neq 1$. Without loss of generality, we may assume $j\neq 1.$ By (3.14) and the concavity of $f$ from [6], we have

[TABLE]

Comparing the coefficients of $|f_{ii}|$ in the last inequality with $a_{ii}(f),$ we have

[TABLE]

Combining the concavity of $f,$ (3.10), (3.16), (3.17), and (3.19), we have at $p,$

[TABLE]

Therefore, we complete the proof of (3.11), with $C=2.1+1.2(n-1)$. ∎

Remark 3.2**.**

In the above proof, we can fix a sufficiently small constant $\varepsilon_{0}$ independent of $\mu,$ and then take $r_{1_{\mu}}=r_{0_{\mu}}\varepsilon_{0}.$ Hence, $r_{1_{\mu}}$ depends on $\mu$ continuously as $r_{0_{\mu}}$ does, which is drawn from [7].

Now we are ready to prove the refined expansions.

Proof of Theorem 1.2.

Fix any point $x\in\Omega$ close to $x_{0}$, with $\text{dist}(x,\partial\Omega)\geq\delta|x-x_{0}|.$ Set

[TABLE]

and

[TABLE]

Let $\widetilde{f}$, $\widehat{f}$, $f_{*}$ be the solutions of (1.1) for $\Omega=\widetilde{\Omega}$, $\widehat{\Omega}$, $\Omega_{x_{0},\mu,\kappa_{1},\kappa_{2}}$, respectively. Then,

[TABLE]

and

[TABLE]

Write $\widehat{p}=x+(1-|x-x_{0}|^{1+\alpha-\epsilon})^{-1}(x_{0}-x)$. For $|x-x_{0}|$ small, it is straightforward to verify

[TABLE]

and

[TABLE]

where $C_{0}$ is some constant depending only on $R$, $\mu$, $\alpha$, $\epsilon$, $\delta$, $h$ and the $C^{2,\alpha}$-norms of $\sigma_{1}$ and $\sigma_{2}$ in $B_{R}(x_{0})$.

Let $f^{\prime}$, $\widehat{f}^{\prime}$ be the solutions of (1.1) on $\widetilde{\Omega}^{\prime}$ and $\widehat{\Omega}^{\prime},$ respectively. We choose $2+\beta=\frac{(2+\alpha)(1+\alpha-\epsilon)}{\epsilon}$ in Lemma 3.1. Then,

[TABLE]

and

[TABLE]

where we took $R_{0}=C_{0}|x-x_{0}|^{\frac{2+\alpha-\epsilon}{2+\alpha}}$ in (3.1). By the maximum principle, we have

[TABLE]

Hence,

[TABLE]

Therefore,

[TABLE]

This completes the proof. ∎