Everywhere differentiability of absolute minimizers for locally strongly convex and concave Hamiltonian $H(p)\in C^0(\mathbb{R}^n)$ with $n\ge3$

TL;DR

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Contribution

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Abstract

Suppose that $n\ge3$ and $H(p)\in C^0(\mathbb{R}^n)$ is a locally strongly convex and concave Hamiltonian. We obtain the everywhere differentiability of all absolute minimizers for $H$ in any domain of $\mathbb{R}^n$.

Equations

$${\mathcal{F}}_{H}\left(u,\Omega\right)=\mathop{\mathrm{missing}}{\,esssup\,}_{x\in\Omega}H\left(Du(x)\right)\ \mbox{for any domain $\Omega\subset{{{\mathbb{R}}}^{n}}$ and function $u\in W^{1,\infty}_{\mathop{\mathrm{missing}}{\,loc\,}}\left(\Omega\right)$};$$

$${\mathcal{F}}_{H}\left(u,V\right)\leq{\mathcal{F}}_{H}\left(v,V\right)\ \mbox{whenever $V\Subset\Omega$, $v\in W^{1,\infty}_{{\mathop{\mathrm{missing}}{\,loc\,}}}\left(V\right)\cap C(\overline{V})$ and $u=v$ on $\partial V$}.$$

$${\mathcal{A}}_H(u):=\sum _{i,j=1}^n{H}_{p_i}\left(Du\right)H_{p_j}\left(Du\right)u_{x_i{x}_j}=0\text{mbox}\mathrm{in}\Omega ,$$

$${\Delta }_{\infty }u:=\sum _{i,j=1}^n{u}_{x_i}{u}_{x_j}{u}_{x_i{x}_j}=0\text{mbox}\mathrm{in}\Omega$$

$$H(p)-\frac{\lambda}{2}|p|^{2}\quad\mbox{ (resp. }\ \frac{\Lambda|p|^{2}}{2}-H(p)\mbox{) is convex in $U$}.$$

$$H(e)=\|H(Du)\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}\quad{\rm and}\quad\lim_{R\to\infty}\frac{1}{R}u(Rx)=e\cdot x\quad\mbox{ locally uniformly in ${{{\mathbb{R}}}^{n}}$}.$$

$$H(p)-\frac{\lambda }{2}|p{|}^2$$

$$\underset{k\to \infty }{\lim }\underset{y\in B\left(0,1\right)}{\sup }\left|\frac{u(x+r_{j_k}y)-u\left(x\right)}{r_{j_k}}-e_{\{r_{j_k}{\}}_{k\in \mathbb{N}}}\cdot y\right|=0$$

$$H(e_{\{r_{j_k}{\}}_{k\in \mathbb{N}}})=\underset{r\to 0}{\lim }\Vert H\left(Du\right){\Vert }_{L^{\infty }\left(B\left(x,r\right)\right)}.$$

$$H^{\gamma }(p)={\tilde{H}}^{\gamma }(p+p^{\gamma })-{\tilde{H}}^{\gamma }(p^{\gamma })\forall p\in {\mathbb{R}}^n.$$

$$u^{\gamma }\in A{M}_{H^{\gamma }}(U)\cap C^0(\overline{U})$$

$$\Vert H^{\gamma }(D{u}^{\gamma }){\Vert }_{L^{\infty }(U)}\le C\Lambda \Vert u^{\gamma }{\Vert }_{C^{0,1}(\partial U)}\forall \gamma \in (0,1],$$

$${\mathcal{A}}_{H^{\gamma }}(v)+ϵ\Delta v=0\mathrm{in}U;v=u^{\gamma }\mathrm{on}\partial U.$$

$$\Vert u^{\gamma ,ϵ}{\Vert }_{C^0(\overline{U})}\le \Vert u^{\gamma }{\Vert }_{C^0(\partial U)}\forall ϵ\in (0,1].$$

$$\|Du^{{\gamma},\epsilon}\|_{C^{0}(\overline{V})}\leq C_{0}(\lambda,\Lambda,{\rm dist}(V,\partial U),\|u^{{\gamma}}\|_{C^{0}(\partial U)})\quad\mbox{ for any $V\Subset U$ and $\epsilon\in(0,1]$,}$$

$$|u^{\gamma ,ϵ}(x)-u^{\gamma }(x_0)|\le C_{\ast }|x-x_0|\forall x\in U,x_0\in \partial U.$$

$$-H_{p_i}(D{u}^{\gamma ,ϵ})H_{p_j}(D{u}^{\gamma ,ϵ})v_{x_i{x}_j}-2H_{p_i{p}_l}(D{u}^{\gamma ,ϵ})u_{x_i{x}_j}^{\gamma ,ϵ}{H}_{p_j}(D{u}^{\gamma ,ϵ})v_{x_l}-ϵ\Delta v.$$

$$\underset{B(0,3)}{\max }|u^{\gamma ,ϵ}-x_n|\le \tau$$

$$H^{\gamma }(D{u}^{\gamma ,ϵ}(x_0))\ge H^{\gamma }(e_n)-\delta$$

$$|D{u}^{\gamma ,ϵ}(x_0)-e_n{|}^2$$

$$C_1(\lambda ,\Lambda ){\tau }^{1/2}\le \frac{{\theta }^2}{32},\forall \delta <{\delta }_{\theta },\tau <{\tau }_{\theta }.$$

$$\max_{B(0,3)}{|u_{j}(x)-x_{n}|}=\max_{B(0,3r_{j})}\frac{|u(x)-a\cdot x|}{r_{j}}\rightarrow 0\quad\mbox{as $j\to\infty$},$$

$$\underset{B(0,3)}{\max }|u_j(x)-x_n|<\frac{\tau }{4}\forall j\ge j_{\tau }.$$

$$u_j^{\gamma }\in A{M}_{H^{\gamma }}(B(0,3))$$

$$\underset{B(0,3)}{\max }|u_j^{\gamma }-x_n|\le \frac{\tau }{2},\forall \gamma \in (0,{\gamma }_{j,\tau }].$$

$$\underset{B(0,3)}{\max }|u_j^{\gamma ,ϵ}-x_n|\le \tau \forall ϵ<{ϵ}_{\gamma ,j,\tau }.$$

$$\max_{B(0,s_{k}/r_{j})}\frac{|u_{j}(x)-b\cdot x|}{s_{k}/r_{j}}=\max_{B(0,s_{k})}\frac{|u(x)-b\cdot x|}{s_{k}}\rightarrow 0\quad\mbox{ as $k\to\infty$}$$

$$\underset{B(0,s_k/r_j)}{\max }\frac{|u_j-b\cdot x|}{s_k/r_j}\le \frac{\eta }{4}.$$

$$\underset{B(0,s_k/r_j)}{\max }\frac{|u_j^{\gamma }-b\cdot x|}{s_k/r_j}\le \frac{\eta }{2}.$$

$$\underset{B(0,s_k/r_j)}{\max }\frac{|u_j^{\gamma ,ϵ}-b\cdot x|}{s_k/r_j}\le \eta .$$

Full text

Everywhere differentiability of

absolute minimizers

for locally strongly convex and concave

Hamiltonian $H(p)\in C^{0}({{{\mathbb{R}}}^{n}})$ with $n\geq 3$

Peng Fa, Qianyun Miao and Yuan Zhou

Department of Mathematics, Beihang University, Beijing 100191, P. R. China

pengfa@buaa.edu.cn

School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

qianyunm@math.pku.edu.cn

Department of Mathematics, Beihang University, Beijing 100191, P. R. China

yuanzhou@buaa.edu.cn

Abstract. Suppose that $n\geq 3$ and $H(p)\in C^{0}({\mathbb{R}}^{n})$ is a locally strongly convex and concave Hamiltonian. We obtain the everywhere differentiability of all absolute minimizers for $H$ in any domain of ${{{\mathbb{R}}}^{n}}$.

1. Introduction

Let $n\geq 2$ and suppose that $H\in C^{0}({{{\mathbb{R}}}^{n}})$ is convex and coercive (i.e., $\liminf_{p\to\infty}H(p)=\infty$). Aronsson 1960’s initiated the study of minimization problems for the $L^{\infty}$-functional

[TABLE]

see [2, 3, 4, 5]. Given any domain $\Omega\subset{{{\mathbb{R}}}^{n}}$, by Aronsson a function $u\in W^{1,\infty}_{{\mathop{\mathrm{missing}}{\,loc\,}}}\left(\Omega\right)$ is called an absolute minimizer for $H$ in $\Omega$ (write $u\in AM_{H}\left(\Omega\right)$ for simplicity) if

[TABLE]

It turns out that the absolute minimizer is the correct notion of minimizers for such $L^{\infty}$-functionals.

The existence of absolute minimizers for given continuous boundary in bounded domains was proved by Aronsson [4] for $\frac{1}{2}|p|^{2}$ and Barron-Jensen-Wang [9] for general $H(p)\in C^{0}({{{\mathbb{R}}}^{n}})$; while their uniqueness was built up by Jensen [26] for $\frac{1}{2}|p|^{2}$ (see also [1, 8, 13]), and by Jensen-Wang-Yu [27] and Armstrong-Crandal-Julin-Smart [7] for $H(p)\in C^{2}({{{\mathbb{R}}}^{n}})$ and $H(p)\in C^{0}({{{\mathbb{R}}}^{n}})$, respectively, with $H^{-1}(\min H)$ having empty interior.

Moreover, if $H\in C^{1}({{{\mathbb{R}}}^{n}})$ is convex and coercive, absolute minimizers coincide with viscosity solutions to the Aronsson equation (a highly degenerate nonlinear elliptic equation)

[TABLE]

see Jensen [26] for $H(p)=\frac{1}{2}|p|^{2}$, and Crandall-Wang-Yu [15] and Yu [33] (and also [7, 9, 10, 23, 13]) in general. Here $H_{p_{i}}=\frac{\partial H}{\partial p_{i}}$ for $H\in C^{1}({{{\mathbb{R}}}^{n}})$, $u_{x_{i}}=\frac{\partial u}{\partial x_{i}}$ for $u\in C^{1}({{{\mathbb{R}}}^{n}})$, and $u_{x_{i}x_{j}}=\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}$ for $u\in C^{2}({{{\mathbb{R}}}^{n}})$. For the theory of viscosity solution see [14]. In the special case $H(p)=\frac{1}{2}|p|^{2}$, the Aronsson equation (1.1) is the $\infty$-Laplace equaiton

[TABLE]

and its viscosity solutions are called as $\infty$-harmonic functions. If $H\in C^{0}({{{\mathbb{R}}}^{n}})$ but $\not\in C^{1}({{{\mathbb{R}}}^{n}})$, we refer to [13, 7] for further discussions and related problems on the Euler-Lagrange equation for absolute minimizers.

The regularity of absolute minimizer is then the main issue in this field.

By Aronsson [6], $\infty$-harmonic functions are not necessarily $C^{2}$-regular; indeed $\infty$-harmonic function $x_{1}^{4/3}-x_{2}^{4/3}$ in whole ${{{\mathbb{R}}}^{n}}$ is not $C^{2}$-regular. Such a function also leads to a well-known conjecture on the $C^{1,1/3}$- and $W^{2,t}_{\mathop{\mathrm{missing}}{\,loc\,}}$-regularity with $1\leq t<3/2$ of $\infty$-harmonic functions. A seminar step towards this is made by Crandall-Evans [11], who obtained their linear approximation property. They [12] also proved that all bounded $\infty$-harmonic functions in whole ${{{\mathbb{R}}}^{n}}$ with $n\geq 2$ must be constant functions.

Next, when $n=2$, Savin [30] established their interior $C^{1}$-regularity and then deduced the corresponding Liouville theorem, that is, all $\infty$-harmonic functions in whole plane with a linear growth at $\infty$ (that is, $|u(x)|\leq C(1+|x|)$ for all $x\in{\mathbb{R}}^{2}$) must be linear functions. Later, the interior $C^{1,\alpha}$-regularity for some $0<\alpha\ll 1/3$ was proved by Evans-Savin [17] and the boundary $C^{1}$-regularity by Wang-Yu [32]. Recently, Koch-Zhang-Zhou [28] proved that $|Du|^{\alpha}\in W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ for all $\alpha>0$ and all $\infty$-harmonic functions $u$ in planar domains, which is sharp as $\alpha\to 0$; also that the distributional determinant $-\det D^{2}u\,dx$ is a nonnegative Radon measure.

Moreover, when $n\geq 3$, Evans-Smart [18, 19] obtained their everywhere differentiability; Miao-Wang-Zhou [29] and Hong-Zhao [25] independently observed an asymptotic Liouville property, that is, if $u$ is a $\infty$-harmonic function in whole ${{{\mathbb{R}}}^{n}}$ with a linear growth at $\infty$, then $\lim_{R\to\infty}\frac{1}{R}u(Rx)=e\cdot x$ locally uniformly for some vector $e$ with $|e|=\|Du\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}$. But $C^{1},C^{1,\alpha}$-regularity and the corresponding Liouville theorem of $\infty$-harmonic functions are completely open.

On the other hand, if $H\in C^{2}({{{\mathbb{R}}}^{n}})$ is locally strongly convex, Wang-Yu [31] obtained the linear approximation property of absolute minimizer, and when $n=2$, the $C^{1}$-regularity and hence the corresponding Liouville theorem. In this paper, we say that $H\in C^{0}({\mathbb{R}}^{n})$ is locally strongly convex (resp. concave) if for any convex subset $U$ of ${{{\mathbb{R}}}^{n}}$, there exists $\lambda>0$ depending on $U$ (resp. $\Lambda>0$) such that

[TABLE]

Note that $H\in C^{2}({{{\mathbb{R}}}^{n}})$ implies that $H$ is always locally strongly concave. In particular, the $l_{\alpha}$-norm for $2<\alpha<\infty$ provides a class of typical example of locally strongly convex and concave but non-Hilbertian Hamiltonians.

Recently, under the assumptions that $H\in C^{0}({{{\mathbb{R}}}^{n}})$ is convex and coercive, it was shown by Fa-Wang-Zhou [20] that $H$ is not a constant in any line segment if and only if all absolute minimizers for $H$ have the linear approximation property; moreover, when $n=2$, if and only if all absolute minimizers for $H$ are $C^{1}$-regular, and also if and only if the corresponding Liouville theorem holds. In [21], we proved that if $H\in C^{2}({\mathbb{R}}^{2})$ is locally strongly convex and concave, then $H(Du)^{\alpha}\in W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ for all $\alpha>\frac{1}{2}-\tau_{H}$ for all absolute minimizers $u$ in planar domains, where $0<\tau_{H}\leq\frac{1}{2}$ and $\tau_{H}=1/2$ when $H\in C^{2}({\mathbb{R}}^{2})$; and also that the distributional determinant $-\det D^{2}u\,dx$ is a nonnegative Radon measure. But, when $n\geq 3$, the everywhere differentiability, $C^{1},C^{1,\alpha}$-regularity and the Liouville theorem is not clear.

If $n\geq 3$ and $H\in C^{0}({\mathbb{R}}^{n})$ is locally strongly convex and concave, this paper aims to prove the following everywhere differentiability (Theorem 1.1 below) and asymptotic Liouville property (Theorem 1.2 below) of absolute minimizers.

Theorem 1.1**.**

Suppose that $n\geq 3$ and $H\in C^{0}(\mathbb{R}^{n})$ is locally strongly convex and concave. Let $\Omega\subset{{{\mathbb{R}}}^{n}}$ be any domain. If $u\in AM_{H}(\Omega)$, then $u$ is differentiable everywhere in $\Omega$.

Theorem 1.2**.**

Suppose that $n\geq 3$ and $H\in C^{0}(\mathbb{R}^{n})$ is locally strongly convex/concave. If $u\in AM_{H}({{{\mathbb{R}}}^{n}})$ with a linear growth at $\infty$, then there exists a unique vector $e$ such that

[TABLE]

When $n\geq 3$, it is unclear to us whether the assumption for $H$ in Theorems 1.1&1.2 can be relaxed to the weaker (and also necessary in some sense) assumption that $H\in C^{0}({{{\mathbb{R}}}^{n}})$ is convex and coercive and is not a constant in any line segment. By [20], if $H\in C^{0}({{{\mathbb{R}}}^{n}})$ is convex and coercive, and is constant in some line-segment, then both of Theorems 1.1&1.2 are not necessarily true.

In particular, it would be interesting to prove the everywhere differentiability of absolute minimizer for $l_{\alpha}$-norm with $1<\alpha<2$. Recall that if $2<\alpha<\infty$, then $l_{\alpha}$-norm belongs to $C^{2}({{{\mathbb{R}}}^{n}})$ and is convex, and hence both of the conclusions of Theorem 1.1&1.2 holds. If $\alpha=1$ or $\infty$, the $l_{\alpha}$-norm will be constant in some line-segment.

By Remark 1.3 below, we only need to prove Theorems 1.1&1.2 when $H\in C^{0}({{{\mathbb{R}}}^{n}})$ satisfies

1. (H1)

$H$ is strongly convex and concave in ${\mathbb{R}}^{n}$, that is, there exist $0<\lambda\leq\Lambda<\infty$ such that

[TABLE] 2. (H2)

$H(0)=\min_{p\in{{{\mathbb{R}}}^{n}}}H(p)=0$.

Remark 1.3**.**

Suppose that $H\in C^{0}(\mathbb{R}^{n})$ is locally strongly convex and concave.

(i) If $u\in AM_{H}(\Omega)$ for some domain $\Omega\subset{{{\mathbb{R}}}^{n}}$, letting $U\Subset\Omega$ be arbitrary subdomain, we have $k=\|Du\|_{L^{\infty}(U)}<\infty$. Next, by [21, Lemma A.8], there exists a $\widetilde{H}$ which is strongly convex/concave in ${\mathbb{R}}^{n}$ and $\widetilde{H}=H$ in $B(0,k+1)$. Thus $u\in AM_{\widetilde{H}}(U)$. The strongly convexity of $\widetilde{H}$ implies that there exists a $p_{0}\in{{{\mathbb{R}}}^{n}}$ such that $\min_{p\in{{{\mathbb{R}}}^{n}}}\widetilde{H}(p)=H(p_{0})$. Set $\bar{H}(p)=\widetilde{H}(p+p_{0})-\widetilde{H}(p_{0})$ for $p\in{{{\mathbb{R}}}^{n}}$. Then $\bar{H}$ satisfies (H1)&(H2). Write $\bar{u}(x)=u(x)-p_{0}\cdot x$ for all $x\in U$. We have $\bar{u}\in AM_{\bar{H}}(U)$. Since $u$ and $\bar{u}$ have the same regularity in $U$, we only need to prove the everywhere differentiability of $\bar{u}$ in $U$.

(ii) If $u\in AM_{H}({{{\mathbb{R}}}^{n}})$ has a linear growth at $\infty$, then by [20] we have $k:=\|Du\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}<\infty$. Let $\bar{u}$ and $\bar{H}$ as above. Then $u$ is linear if and only if $\bar{u}$ is linear. So we only need to prove $\bar{u}$ is linear.

Unless other specifying, we always assume that $H\in C^{0}({{{\mathbb{R}}}^{n}})$ satisfies (H1)&(H2) below. Note that the geometric&variational approach used in dimension 2 (see Savin [30] and also [20, 31]) is not enough to prove Theorems 1.1&1.2, since it includes a key planar topological argument. Moreover, since $H\in C^{0}({{{\mathbb{R}}}^{n}})$ does not have Hilbert structure necessarily, it is not clear whether one can prove Theorem 1.1 by using the idea of Evans-Smart [18]—a PDE approach based on maximal principle (see also Remark 2.6 (ii)). But, in Section 2, we are able to prove Theorems 1.1&1.2 by borrowing some idea of Evans-Smart [19]—a PDE approach based on an adjoint argument, and using the following crucial ingredients:

1. (a)

the linear approximation property of any given absolute minimizer $u$ for $H$ as obtained in Fa-Wang-Zhou [20] and Wang-Yu [31] (see Lemmas 2.1&2.5). 2. (b)

a stability result in [21] (see Lemma 2.2) which allows to approximate $u$ via absolute minimizers $u^{\gamma}$ of a Hamiltonian $H^{\gamma}$, where $H^{\gamma}$ is a smooth approximation of $H$ and satisfies (H1)&(H2) with the same constants $\lambda,\Lambda$. 3. (c)

a uniform approximation to $u^{\gamma}$ via smooth functions $u^{{\gamma},\epsilon}$ (see Theorem 2.3), which is an appropriate modification of Evans’ approximation via $e^{\frac{1}{\epsilon}H^{\gamma}}$-harmonic functions in [16]. The point is that none of $k\geq 3$ -order derivatives of $H^{\gamma}$ is involved in the linearization of the equation (2.2) for $u^{{\gamma},\epsilon}$. 4. (d)

an integral flatness estimate for $u^{{\gamma},\epsilon}$ (see Theorem 2.4).

Theorem 2.3 will be proved in Section 3. The novelty in the proof of Theorem 2.3 is that we use viscosity solutions to certain Hamilton-Jacobi equation as barrier functions to get a boundary regularity of $u^{{\gamma},\epsilon}$ and then conclude the uniform approximation of $u^{{\gamma},\epsilon}$ to $u^{\gamma}$. The reason to use $u^{{\gamma},\epsilon}$ instead of $e^{\frac{1}{\epsilon}H^{\gamma}}$-harmonic functions is that the linearization of $e^{\frac{1}{\epsilon}H^{\gamma}}$-harmonic equation contains $3$-order derivatives of $H^{\gamma}$; see Remark 2.6 (i) for details.

Theorem 2.4 will be proved in Section 5. To this end, we generalize in Section 4 the adjoint arguments of [19] to Hamiltonian $H^{\gamma}$ and $u^{{\gamma},\epsilon}$. Since none of $k\geq 3$ -order derivatives of $H^{\gamma}$ is involved in the equation for $u^{{\gamma},\epsilon}$, all key estimates in Theorem 2.3 and Section 4 rely only on $\lambda$ and $\Lambda$. This is indeed important to get Theorem 2.4. Moreover, since $H\in C^{0}({{{\mathbb{R}}}^{n}})$ does not have Hilbert structure in general, some new ideas are needed to get Theorem 2.4 in Section 5; in particular, the test function used in the proof of flatness estimates in [19] is not enough to us, as an another novelty we find a suitable test function and build up some related estimates.

2. Proofs of Theorems 1.1&1.2

Considering Remark 1.3, we always assume that $H\in C^{0}({{{\mathbb{R}}}^{n}})$ satisfies (H1)&(H2). To prove Theorem 1.1, let $\Omega$ be any domain of ${{{\mathbb{R}}}^{n}}$, and $u\in AM_{H}(\Omega)$. We recall the following linear approximation property of $u$ as established by [20].

Lemma 2.1**.**

For any $x\in\Omega$ and any sequence $\{r_{j}\}_{j\in{\mathbb{N}}}$ which converges to [math], there exist a subsequence $\{r_{j_{k}}\}_{k\in{\mathbb{N}}}$ and a vector $e_{\{r_{j_{k}}\}_{k\in{\mathbb{N}}}}$ such that

[TABLE]

and

[TABLE]

For each $x\in\Omega,$ denote by $\mathscr{D}u(x)$ the collection of all possible vector $e_{\{r_{j_{k}}\}_{k\in{\mathbb{N}}}}$ as above. Observe that $u$ is differentiable at $x$ if and only if $\mathscr{D}u(x)$ is a singleton; in this case $\mathscr{D}u(x)=\{Du(x)\}$.

To see that $\mathscr{D}u(x)$ is a singleton, we need the following approximation to $u$ given in [21]. Precisely, let $\{H^{\gamma}\}_{\gamma\in(0,1]}$ be a standard smooth approximation to $H$ as below. For each ${\gamma}\in(0,1]$, let $\widetilde{H}^{\gamma}=\eta_{\gamma}\ast H$, where $\eta_{\gamma}$ is standard smooth mollifier. Since $H^{\gamma}$ is strictly convex there exists a unique point $p^{\gamma}\in{\mathbb{R}}^{2}$ such that $\widetilde{H}^{\gamma}(p^{\gamma})=\min_{p\in{\mathbb{R}}^{2}}\widetilde{H}^{\gamma}(p).$ Set

[TABLE]

Obviously, $H^{\gamma}$ satisfies (H2); by [21, Appendix A], $\{H^{\gamma}\}_{\gamma\in(0,1]}$ satisfies (H1) with the same $\lambda$ and $\Lambda$, and $H^{\gamma}\to H$ locally uniformly as ${\gamma}\to 0$. For each ${\gamma}\in(0,1]$ and $U\Subset\Omega$, let

[TABLE]

We then have the following result; see [21] for $n=2$ and note that the proofs in [21] also works for $n\geq 3$.

Lemma 2.2**.**

We have

[TABLE]

and $u^{\gamma}\to u$ in $C^{0}(U)$ as ${\gamma}\to 0$.

Next, for any ${\gamma}\in(0,1]$, to approximate $u^{\gamma}$ in a smooth way we consider the following Dirichlet problem:

[TABLE]

The following result is proved in Section 3.

Theorem 2.3**.**

For each $\epsilon,{\gamma}\in(0,1]$, there exists a unique solution $u^{{\gamma},\epsilon}\in C^{\infty}(U)\cap C^{0}(\overline{U})$ to (2.2). Moreover, the following hold.

1. (i)

We have

[TABLE] 2. (ii)

We have

[TABLE]

where the constant $C_{0}(\lambda,\Lambda,{\rm dist}(V,\partial U),\|u^{{\gamma}}\|_{C^{0}(\partial U)})$ depends only on $\lambda$, $\Lambda$, ${\rm dist}(V,\partial U)$ and $\|u^{{\gamma}}\|_{C^{0}(\partial U)}$. 3. (iii)

There exist $\epsilon_{\ast}>0$ and $C_{\ast}>0$ depending on $H^{\gamma}$ and $\|u^{\gamma}\|_{C^{0,1}(\overline{U})}$ such that for any $\epsilon\in(0,\epsilon_{\ast})$, we have

[TABLE] 4. (iv)

We have $u^{{\gamma},\epsilon}\to u^{\gamma}$ in $C^{0}(\overline{U})$ as $\epsilon\to 0$.

The existence and uniqueness of $u^{{\gamma},\epsilon}$, and also Theorem 2.3 (i) follow from the classical elliptic theory; Theorem 2.3 (iv) from Theorem 2.3 (ii) and (iii). Theorem 2.3 (ii) follows from the approach by [18] based on the maximal principle and the linearized operator arising from (2.2):

[TABLE]

Since none of $k\geq 3$ order derivatives of $H$ is involved in (2.3), we will conclude that the constant $C_{0}$ in Theorem 2.3 (ii) depends at most on $\lambda$, $\Lambda$ ${\rm dist}(V,\partial U)$ and $\|u^{{\gamma}}\|_{C^{0}(\partial U)}$. To get Theorem 2.3 (iii), we need new ideas. Indeed, unlike the case $H(p)=\frac{1}{2}|p|^{2}$, where we use $|x|^{\gamma}$ as a barrier function to conclude Theorem 2.3 (iii) from the comparison principle, the novelty here is that due to we take viscosity solutions $\mathcal{L}^{\delta}_{\sigma}$ of certain Hamilton-Jacobi equation as barrier functions; see Lemmas 3.1-3.2.

In Section 5, we establish the following flatness estimate of $u^{{\gamma},\epsilon}$, which is is crucial to show that $\mathscr{D}u(x)$ is a singleton. Denote by $e_{n}$ the vector $(0,\cdots,0,1)$.

Theorem 2.4**.**

Suppose that $U=B(0,3)$ and for some ${\gamma},\epsilon\in(0,1]$, $u^{{\gamma},\epsilon}$ satisfies

[TABLE]

for some $0<\tau<1$ and

[TABLE]

for some $0<\delta<H(e_{n})/2$ and $x_{0}\in B(0,1)$. Then

[TABLE]

where $\mu=\frac{\lambda}{16n}$. Above $C_{1}(\lambda,\Lambda)$ is a constant depending only on $\lambda$ and $\Lambda$.

The proof of Theorem 2.4 relies on a generalization of the adjoint method of Evans-Smart [19] to the equation (2.2) as developed in Section 4. Moreover, since $H$ does not have Hilbert structure necessarily, we can not follow the argument of Evans-Smart to get Theorem 2.4, where they take $u^{{\gamma},\epsilon}_{x_{n}}-1$ as a test function. The novelty here is to take $|Du^{{\gamma},\epsilon}-e_{n}|^{2}$ as a test function. With aid of the estimates in Section 4, by using the strongly convexity/concavity of $H$ and some careful analysis, we are able to prove Theorem 2.4. Again, since none of $k\geq 3$ order derivatives of $H$ are involved in the linearized operator and hence in the whole procedure, we conclude that all constants in estimates in Section 4 and hence $C_{1}$ in Theorem 2.4 depend on at most $\lambda$ and $\Lambda$.

With the aid of Theorem 2.4, Theorem 2.3 and Lemma 2.1, by some necessary modifications of the arguments of [18] we are able to prove that for any $x\in\Omega$, $\mathscr{D}u(x)$ is singleton, and hence that $u$ is differentiable everywhere in $\Omega$; for reader’s convenience we give the details.

Proof of Theorem 1.1.

By Remark 1.3, we assume $H\in C^{0}({{{\mathbb{R}}}^{n}})$ satisfies (H1)&(H2). Let $\Omega$ be any domain of ${{{\mathbb{R}}}^{n}}$, and $u\in AM_{H}(\Omega)$. It suffices to prove that $\mathscr{D}u(x)$ is singleton. We prove this by contradiction. Assume that $\mathscr{D}u(x_{0})$ contains at least two vectors $a\neq b$ with $H(a)=H(b)$ for some $x_{0}\in\Omega$. Note that $a,b\neq 0$. We may assume that $x_{0}=0\in\Omega$, $u(0)=0$, $a=e_{n}$ without loss of generality. Set $\theta=|b-e_{n}|>0$. We obtain a contradiction by the following 4 steps.

Step 1. Fix $\tau_{\theta}\in(0,1]$ such that

[TABLE]

Since $e_{n}\in\mathscr{D}u(0)$ we can find a sequence $\{r_{j}\}_{j\in{\mathbb{N}}}$ which converges to [math] such that

[TABLE]

where $u_{j}(x)=u(r_{j}x)/{r_{j}}$ for $x\in\frac{1}{r_{j}}\Omega$. For each $\tau\in(0,\tau_{\theta}]$, there exists a $j_{\tau}$ such that if $j\geq j_{\tau}$,

[TABLE]

For any ${\gamma}\in(0,1)$ and $j\in{\mathbb{N}}$, let

[TABLE]

By Lemma 2.2, for each $j\geq j_{\tau}$, $u_{j}^{\gamma}\to u_{j}$ as ${\gamma}\to 0$, there exists $\gamma_{j,\tau}>0$ such that

[TABLE]

By Theorem 2.3 (iv), for each $j\geq j_{\tau}$ and ${\gamma}<\gamma_{j,\tau}$, there is an $\epsilon_{{\gamma},j,\tau}\in(0,1]$ that

[TABLE]

Step 2. Since $b\in\mathscr{D}(u)(0)$ by [20], there exist a sequence $\{s_{k}\}_{k=1}^{\infty}$ which converge to zero such that

[TABLE]

for all $j\in{\mathbb{N}}$. For each $\eta\in(0,\tau)$ and $j\geq j_{\tau}$, there exist $k_{\eta,j}\in{\mathbb{N}}$ such that for all $k\geq k_{\eta,j}$, we have $s_{k}/r_{j}\leq 1$ and

[TABLE]

Since $u^{\gamma}_{j}\to u_{j}$ in $B(0,3)$, for each $k\geq k_{\eta,j}$ we can find ${\gamma}_{k,j,\eta}<{\gamma}_{j,\tau}$ such that for each ${\gamma}\in(0,{\gamma}_{k,j,\eta})$,

[TABLE]

Since $u^{{\gamma},\epsilon}_{j}\to u^{\gamma}_{j}$ in $B(0,3)$, for each ${\gamma}\in(0,{\gamma}_{k,j,\eta})$, we further find $\epsilon_{{\gamma},k,j,\eta}<\epsilon_{{\gamma},j,\tau}$ such that for all $\epsilon<\epsilon_{{\gamma},k,j,\eta}$,

[TABLE]

Step 3. For each $\eta\in(0,\tau)$, there exists ${\gamma}_{\eta}\in(0,1)$ such that

[TABLE]

where we have used $H(e_{n})=H(b)$.

For each $\eta\in(0,\tau)$, $j\geq j_{\tau}$, $k\geq k_{j,\eta}$, ${\gamma}<\min\{{\gamma}_{k,j,\eta},{\gamma}_{\eta}\}$ and $\epsilon<\epsilon_{{\gamma},k,j,\eta}$, by Lemma [18, 19], (2.11) implies that there is a point $x_{\epsilon,{\gamma},k,j,\eta}\in B(0,s_{k}/r_{j})$ at which

[TABLE]

We further have that

[TABLE]

Indeed, by convexity of $H$, we have

[TABLE]

Since $|D_{p}H^{\gamma}(p)|\leq\Lambda|p|$ and $|Du^{\gamma,\epsilon}_{j}(x^{0})|\leq|b|+4$, one has

[TABLE]

A similar estimate holds for $H^{\gamma}(b)-H^{\gamma}(Du^{\epsilon,\gamma}_{j}(x_{\epsilon,{\gamma},k,j,\eta}))$. Thus

[TABLE]

This together with $|H^{\gamma}(e_{n})-H^{\gamma}(b)|\leq\eta$ implies (2.13).

Step 4. Let $\delta_{\theta}\in(0,1]$ such that

[TABLE]

For each $\mu\in(0,\frac{\lambda}{8n}]$, $\delta\in(0,\delta_{\theta}]$, let $\epsilon_{\mu,\delta,\theta}\in(0,1]$ such that

[TABLE]

Let $\eta<\min\{\delta_{\theta}/C_{2}(\lambda,\Lambda,b),\theta/{16}\}$ and $\delta=C_{2}(\lambda,\Lambda,b)\eta$. For $\tau<\tau_{\theta}$, $j\geq j_{\tau}$, $k\geq k_{j,\eta}$, ${\gamma}<\min\{{\gamma}_{k,j,\eta},{\gamma}_{\eta}\}$, and $\epsilon<\min\{\epsilon_{{\gamma},k,j,\eta},\epsilon_{\mu,\delta,\theta}\}$, by Theorem 2.4, (2.13) and (2.10) imply that

[TABLE]

Thus by (2.12) one has

[TABLE]

which is a contradiction as desired. The proof of Theorem 1.1 is complete. ∎

To prove Theorem 1.2, let $u\in AM_{H}({{{\mathbb{R}}}^{n}})$ with a linear growth at $\infty$. By [20], $\|Du\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}<\infty$ and moreover $u$ has the linear approximation property at $\infty$ as below.

Lemma 2.5**.**

For any sequence $\{r_{j}\}_{j\in{\mathbb{N}}}$ which converges to $\infty$, there exist a subsequence $\{r_{j_{k}}\}_{k\in{\mathbb{N}}}$ and a vector $e_{\{r_{j_{k}}\}_{k\in{\mathbb{N}}}}$ such that

[TABLE]

and

[TABLE]

Denote by $\mathscr{D}(\infty)$ the collection of all possible $e_{\{r_{j_{k}}\}}$ as above. Following the proof of Theorem 1.1 line by line and letting $r_{k}\to\infty$ as $k\to\infty$, we are able to prove that $\mathscr{D}(\infty)$ is singleton, and hence prove Theorem 1.2; here we omit the details and also refer to [25, 29].

We end this section by the following remark.

Remark 2.6**.**

(i) Recall that Evans [16] suggested another approximation to $u^{\gamma}$ via $e^{\frac{1}{\epsilon}H}$-harmonic functions $\hat{u}^{{\gamma},\epsilon}$, that is, smooth solutions to

[TABLE]

But note that the 3-order derivative of $H^{\gamma}$ appears in third terms of the linearized operator

[TABLE]

If we want to get Theorem 2.3 and 2.4 for $\hat{u}^{{\gamma},\epsilon}$ so that the constants $C_{0},C_{1}$ are independent of $3$-order derivative of $H^{\gamma}$ or $H$, some extra efforts are needed. To avoid such extra efforts, we prefer to consider the approximation equation (2.2).

(ii) If $H(p)=\frac{1}{2}|p|^{2}$, a flatness estimate stronger than Theorem 2.4 is also given in [18] via the maximal principle,

[TABLE]

Note that in this case, $H^{\gamma}=H$ and $u^{\gamma}=u$, and $u^{{\gamma},\epsilon}$ is then reduced to $u^{\epsilon}$. From this Evans-Smart [18] concluded the everywhere differentiability of $\infty$-harmonic functions $u$. But for $H\in C^{0}({{{\mathbb{R}}}^{n}})$ satisfying (H1) and (H2), since $H$ does not necessarily have a Hilbert structure, it is still unclear whether there is some estimate similar to (2.18), and also whether the approach in [18] can be used to prove Theorem 1.1.

3. Proof of Theorem 2.3

Let $H$, $H^{\gamma}$, $u$ $u^{\gamma}$ and $u^{{\gamma},\epsilon}$ be as in Section 2. Note that $H^{\gamma}$ satisfies (H1)&(H2) with the same $\lambda$ and $\Lambda$. Since $|H^{\gamma}_{p}(p)|^{2}\leq\Lambda^{2}|p|^{2}$ implies

[TABLE]

by a standard quasilinear elliptic theory (see [24]), there exists a unique smooth solution $u^{{\gamma},\epsilon}\in C^{\infty}(U)\cap C^{0}(\overline{U})$ to (2.2). Theorem 2.3 (i) follows from the known maximum principle. We also note that by a standard argument, $u^{{\gamma},\epsilon}\to u^{\gamma}$ in $C^{0}(\overline{U})$ (that is Theorem 2.3 (iv)) follows from Theorem 2.3 (ii)&(iii), and the uniqueness of $u^{\gamma}$ in [7]; here we omit the details. Below we only need to prove Theorem 2.3 (ii)&(iii). For simplicity, we write $H^{\gamma}$ as $H$, $u^{\gamma}$ as $u$, and we write $u^{{\gamma},\epsilon}$ as $u^{\epsilon}$ by abuse of notation.

We prove Theorem 2.3 (ii) using the approach of Evans-Smart [19] here. Denote by $L_{\epsilon}$ the linearized operator obtained from $\mathscr{A}_{H}[u^{\epsilon}]+\epsilon\Delta u^{\epsilon}=0$, that is,

[TABLE]

for $v\in C^{\infty}(U)$. Note that

[TABLE]

Proof of Theorem 2.3 (ii).

We choose $\zeta\in C^{\infty}_{c}(U)$ such that

[TABLE]

Define an auxiliary function

[TABLE]

where $\alpha>0$ will be determined later. If $w$ attains its maximum on $\partial U$, then

[TABLE]

this implies Theorem 2.3 (ii).

Assume that $w$ attains its maximum at some $x_{0}\in U$. Since $Dw(x_{0})=0$ and $D^{2}w(x_{0})$ is nonpositive definite, we have $L_{\epsilon}(w)\geq 0$ at $x_{0}$. Below we estimate $L_{\epsilon}(w)$ at $x_{0}$ from above. Note that

[TABLE]

A direct calculation gives

[TABLE]

By $L_{\epsilon}(u^{\epsilon}_{x_{k}})=0$, and $D^{2}u^{\epsilon}D_{p}H(Du^{\epsilon})=D[H(Du^{\epsilon})]$ we obtain

[TABLE]

Similarly using (2.2), we have

[TABLE]

Since (H1)&(H2) implies

[TABLE]

by Young’s inequality, we obtain

[TABLE]

Since

[TABLE]

using (H1)&(H2) and Young’s inequality we also obtain

[TABLE]

Similarly,

[TABLE]

In conclusion, we have

[TABLE]

At $x_{0}$, $L_{\epsilon}(w)\geq 0$ implies that

[TABLE]

Multiplying the above inequality with $\zeta^{4}$ yields

[TABLE]

By Young’s inequality we have

[TABLE]

and hence

[TABLE]

Choosing $\alpha=\alpha(\lambda,\Lambda,\|u^{\epsilon}\|_{C^{0}(U)},{\mathop{\mathrm{missing}}{\,dist\,}}(V,\partial U))$ so that

[TABLE]

we have

[TABLE]

Hence,

[TABLE]

as desired. ∎

To prove Theorem 2.3 (iii), we need the following Lemma 3.1, which can be found in [22, Lemma 3.2 and Lemma 3.4]. For each $t>0$, $\delta>0$, $\sigma>0$ and $x,y\in U$, define

[TABLE]

where $\mathcal{C}(0,t;x,y;U)$ is the set of all rectifiable curves $\xi:[0,t]\rightarrow U$ that joins $x$ to $y$, and

[TABLE]

For each $\sigma>0$, we also need to the notion of generalized cones, that is

[TABLE]

By the strongly convexity of $H$, one always has that

[TABLE]

Lemma 3.1**.**

Assume that $H\in C^{\infty}({\mathbb{R}}^{n})$ satisfy (H1)&(H2).

1. (i)

For all $\sigma>0$,$\delta\geq 0$* and $x,y\in U$, we have*

[TABLE] 2. (ii)

When $\frac{\delta}{\sigma}\mathcal{L}^{\delta}_{\sigma}(x,y)<\ln\sqrt{2}$, we also have

[TABLE] 3. (iii)

For any domain $V\Subset{\mathbb{R}}^{n}$ and $x_{0}\in\overline{V}$, we have $\mathcal{L}^{\delta}_{\sigma}(x_{0},\cdot)$ is a viscosity sup-solution of

[TABLE]

and $\mathcal{L}^{\delta}_{\sigma}(\cdot,x_{0})$ is a viscosity sub-solution of

[TABLE]

We also need the following comparison principle, see [33, Appendix, Theorem 2].

Lemma 3.2**.**

Assume that $H\in C^{\infty}({\mathbb{R}}^{n})$ satisfy (H1)&(H2). For any $\sigma>0$ and domain $V\subset{\mathbb{R}}^{n}$, assume that $u_{1}\in C^{0}(\overline{V})$ is a viscosity sup-solution of

[TABLE]

and $u_{2}\in C^{0}(\overline{V})$ is a viscosity sub-solution of

[TABLE]

If either $u_{1}\in C^{0,1}(V)$ or $u_{2}\in C^{0,1}(V)$, then

[TABLE]

From Lemma 3.1 and 3.2, we deduce the following.

Lemma 3.3**.**

Assume that $H\in C^{\infty}({\mathbb{R}}^{n})$ satisfy (H1)&(H2). For any domain $V\Subset{{{\mathbb{R}}}^{n}}$ and $x_{0}\in\overline{V}$ for all $\sigma>0$ and $0<\delta<\delta_{\sigma,V}$, there exist constant $\mu_{1},\mu_{2}>0$ depending on $\sigma,\delta,H$ such that for all $\epsilon\in(0,1)$, $\mathcal{L}^{\lambda}_{\sigma}(x_{0},\cdot)$ is a viscosity sup-solution of

[TABLE]

and $\mathcal{L}^{\lambda}_{\sigma}(\cdot,x_{0})$ is a viscosity sub-solution of

[TABLE]

Proof of Lemma 3.3.

For any $\phi\in C^{2}(V)$ and $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)-\phi(x)$ attains its locally minimum at $y\in V\backslash\{x_{0}\}$, it suffice to prove that

[TABLE]

Without loss of generality, we may assume that $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)-\phi$ attains its a strictly minimum at $y_{0}\in V\backslash\{x_{0}\}$. Since $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)$ is semiconcave, for any $\eta>0$, $r>0$, by Lemma A.3 in [14] there exist $x^{r,\eta}\in B(y_{0},r)$ and $p^{r,\eta}\in B(0,\eta)$ such that $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)-\phi(x)-\langle p^{r,\eta},x\rangle$ has a local minimal at $x^{r,\eta}$ and $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)$ is twice differentiable at $x^{r,\eta}$. Also, the semiconcave property of $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)$ implies that there exists $\mu_{1}>0$ depending on $\sigma,\delta,H$ such that

[TABLE]

in the sense of distributions, where $I_{n}$ is identity matrix. Since $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)$ is twice differentiable at $x^{r,\eta}$, by Lemma 3.1 and (3.3), we have

[TABLE]

On the other hand, since $\mathcal{L}^{\delta}_{\sigma}(x_{0},x)-\phi(x)-\langle p^{r,\eta},x\rangle$ has a local minimal at $x^{r,\eta}$, we have $D\mathcal{L}^{\delta}_{\sigma}(x_{0},x^{r,\eta})=D\phi(x^{r,\eta})+p^{r,\eta}$ and $D^{2}\mathcal{L}^{\delta}_{\sigma}(x_{0},x^{r,\eta})\geq D^{2}\phi(x^{r,\eta})$. Thus

[TABLE]

Combing (3.4) and (3.5), we have

[TABLE]

Letting $r=\eta\to 0$ and noting $p^{r,\eta}\to 0$, $x^{r,\eta}\to y_{0}$, this leads to the (3.2).

Similarly, we can prove that $-\mathcal{L}^{\delta}_{\sigma}(x,x_{0})$ is viscosity sub-solution of

[TABLE]

The proof is complete. ∎

We are able to prove Theorem 2.2 (iii) as below.

Proof of Theorem 2.2 (iii)..

Note that $u\in C^{0,1}(\overline{U})$. Letting $\sigma>8\Lambda\|u\|^{2}_{C^{0,1}(\overline{U})}$, we have

[TABLE]

Moreover, there exist $\delta(\sigma,U)>0$ such that for all $x,y\in U$ and $\delta<\delta(\sigma,U)$, we have

[TABLE]

and hence, by Lemma 3.1,

[TABLE]

By (3.7), for all $\sigma>8\Lambda\|u\|^{2}_{C^{0,1}(\overline{U})}$ and $\delta<\delta(\sigma,U)$, we have

[TABLE]

Note that

[TABLE]

in viscosity sense and by Lemma 3.3,

[TABLE]

in viscosity sense. For all $\sigma>8\Lambda\|u\|^{2}_{C^{0,1}(\overline{U})}$ and $\delta<\delta(\sigma,U)$ and if $0<\epsilon<\frac{\delta\sigma}{2n\mu_{1}}$, by Lemma 3.2 we have

[TABLE]

By similar argument, for all $\sigma>8\Lambda\|u\|^{2}_{C^{0,1}(\overline{U})}$ and $\delta<\delta(\sigma,U)$, if $0<\epsilon<\frac{\delta\sigma}{2n\mu_{2}}$, we have

[TABLE]

We therefore conclude that for $\sigma=8\Lambda\|u\|^{2}_{C^{0,1}(\overline{U})}$ and $\delta<\delta(\sigma,U)$ if $0<\epsilon<\min\{\frac{\delta\sigma}{2n\mu_{1}},\frac{\delta\sigma}{2n\mu_{2}}\}$,

[TABLE]

Thus, there exist $\epsilon_{\ast}$ and $C$ depending on $U$, $\|u\|_{C^{0,1}(\overline{U})}$, $H$, $\delta,\sigma$ such that for all $0<\epsilon<\epsilon_{\ast}$, we have

[TABLE]

The proof of Theorem 2.2 is complete. ∎

4. A generalization of Evans-Smart’ adjoint method

Let $H$, $H^{\gamma}$, $u$ $u^{\gamma}$ and $u^{{\gamma},\epsilon}$ be as in Section 2. For convenience, we write $H^{\gamma}$ as $H$, and $u^{\gamma}$ as $u$, $u^{{\gamma},\epsilon}$ as $u^{\epsilon}$ below. Let $L_{\epsilon}$ be the linearized operator given in (3.1), and $L^{\ast}_{\epsilon}$ be its dual operator, that is,

[TABLE]

for any $v\in C^{\infty}(U)$. Observe that

[TABLE]

Fix a smooth domain $V\Subset U$. For each point $x_{0}\in V$, we consider the adjoint problem

[TABLE]

where $\delta_{x_{0}}$ denotes the Dirac measure at $x_{0}$. Equivalently,

[TABLE]

Then we have the following result.

Theorem 4.1**.**

For each point $x_{0}\in V$, there exists a unique solution $\Theta^{\epsilon}\in C^{\infty}(\overline{V}\setminus\{x_{0}\})$ of the linear adjoint problem (4.2) such that $\Theta^{\epsilon}\geq 0$ in $V$.

Proof.

Consider problem

[TABLE]

By Theorem 2.3, there exists a unique solution $\omega\equiv 0$ on $\overline{V}$. So that [math] is not an eigenvalue of the operator $L_{\epsilon}$, and hence [math] is not an eigenvalue of $L_{\epsilon}^{\star}$. Applying standard linear elliptic PDE theory, there exists smooth Green’s function $\Theta^{\epsilon}\in C^{\infty}(\overline{B}(0,2)\backslash\{x_{0}\})$. Next we show that $\Theta^{\epsilon}\geq 0$. For any $f\in C^{\infty}(V)$ and $f\geq 0$ in $V$, we introduce the solution $\omega^{\epsilon}$ of the linear boundary value problem

[TABLE]

By Theorem 2.3, we know that there exists a unique solution $0\leq\omega^{\epsilon}\in C^{\infty}(V)$. Multiply the equation in (4.3) by $\Theta^{\epsilon}$, we have

[TABLE]

By integration by parts, $\omega^{\epsilon}|_{\partial V}=0$ and $\Theta^{\epsilon}|_{\partial V}=0$, we have

[TABLE]

where $\overrightarrow{N}$ denotes the outward pointing unit normal along $\partial V$. By similar calculation, which lead to

[TABLE]

Since for all $f\geq 0$ holds, that is $\Theta^{\epsilon}\geq 0$. ∎

Lemma 4.2**.**

Denote by $\overrightarrow{N}$ denotes the outward pointing unit normal along $\partial V$. Then

[TABLE]

We have the following connection of between operator $L_{\epsilon}$ and $\Theta_{\epsilon}$.

Lemma 4.3**.**

For any $v\in C^{\infty}(\overline{V})$, we have

[TABLE]

where dentes

[TABLE]

Proof.

By integrate by parts and $\Theta^{\epsilon}|_{\partial V}=0$, we have

[TABLE]

where $\overrightarrow{N}$ denotes the outward pointing unit normal along $\partial V$. Note that $\cos(N,x_{i})=-\Theta^{\epsilon}_{x_{i}}/|D\Theta^{\epsilon}|$ and $\Theta^{\epsilon}|_{\partial V}=0$, we have

[TABLE]

Denote

[TABLE]

this complete proof of the Lemma 4.3. ∎

Since $L_{\epsilon}(1)=0$ in $V$, the following follows from Lemma 4.3 obviously.

Corollary 4.4**.**

We have

[TABLE]

Letting $v=H(Du^{\epsilon})$ in Lemma 4.3, we also have the following.

Lemma 4.5**.**

We have

[TABLE]

Proof.

By Lemma 4.3,

[TABLE]

Write

[TABLE]

Since $L_{\epsilon}(u^{\epsilon}_{x_{s}})=0$, we have

[TABLE]

as desired. ∎

We further need an exponential estimate.

Lemma 4.6**.**

Moreover, for all $0<\mu<\frac{\lambda}{8n}$ we have

[TABLE]

Proof.

Let

[TABLE]

where $0<\mu\leq\frac{\lambda}{8n}$. Similarly to (4.4) we have

[TABLE]

Since

[TABLE]

and $\mathscr{A}_{H}[u^{\epsilon}]=-\epsilon\Delta u^{\epsilon}$, we get

[TABLE]

Note that

[TABLE]

Since the strongly convexity of $H$ implies

[TABLE]

by $[1-\frac{n\mu}{\lambda}]\geq 1/2$ we have

[TABLE]

Since Lemma 4.3 implies

[TABLE]

We obtain the desired estimate. ∎

Applying Lemma 4.6, we will get the following upper bound.

Lemma 4.7**.**

We have

[TABLE]

Proof.

By $\mathscr{A}_{H}(u^{\epsilon})+\epsilon\Delta u^{\epsilon}=0$, a direct calculation implies that

[TABLE]

By the convexity of $H$ and $H(0)=0$, we have

[TABLE]

By the Young’s inequality, we have

[TABLE]

By the strongly concavity of $H$, we have

[TABLE]

Thus

[TABLE]

By Lemma 4.3,

[TABLE]

and hence,

[TABLE]

∎

Moreover, we also need an integral estimate of $\Theta^{\epsilon}$.

Lemma 4.8**.**

Let $x_{0}\in V$ and $\alpha_{\epsilon}:=H(Du^{\epsilon}(x_{0}))>0$.

(i)For any $0<\mu<\frac{\lambda}{8n}$ and $0<\beta<H(Du^{\epsilon}(x_{0}))$, we have

[TABLE]

(ii) If $\liminf_{\epsilon\to 0}H(Du^{\epsilon}(x_{0}))\geq\alpha>0$, we have

[TABLE]

Proof.

For each $0<\mu<\frac{\lambda}{8n}$, define

[TABLE]

and set $v(x)=(\phi\circ H)(Du^{\epsilon}(x))|x|^{2}$. By Lemma 4.3,

[TABLE]

Then

[TABLE]

Write

[TABLE]

By (4.5), we have

[TABLE]

Note that

[TABLE]

and hence by the Young’s inequality,

[TABLE]

Using $\langle D_{p}H(Du^{\epsilon}),D[H(Du^{\epsilon})]\rangle=\mathscr{A}_{H}[u^{\epsilon}]=-\epsilon\Delta u^{\epsilon}$ we also have

[TABLE]

by $\phi^{\prime}(r)=-{\mu}e^{\frac{\mu}{\epsilon}[\alpha_{\epsilon}-H(Du^{\epsilon})]}\leq 0$ and Young’s inequality, which is bounded by

[TABLE]

Thus

[TABLE]

Therefore, applying Lemma 4.6 we get

[TABLE]

We conclude that

[TABLE]

and hence

[TABLE]

This implies that

[TABLE]

Let $C_{0}(\lambda,\Lambda)\eta^{2}\frac{4}{\alpha^{2}}=\frac{1}{2}$, that is, $\eta^{2}=\frac{\alpha^{2}}{8C_{0}(\lambda,\Lambda)}$. Apply Lemma 4.7, Theorem 2.3 and $\alpha_{\epsilon}\geq\alpha$ for all $\epsilon\in(0,1]$, we have

[TABLE]

∎

5. Proof of Theorem 2.4

Let $U=B(0,3)$ and $V=B(0,2)$ in this section. Let $H$, $H^{\gamma}$, $u$ $u^{\gamma}$ and $u^{{\gamma},\epsilon}$ be as in Section 2. For convenience, we write $H^{\gamma}$ as $H$, and $u^{\gamma}$ as $u$, $u^{{\gamma},\epsilon}$ as $u^{\epsilon}$ below.

Note that the condition (2.4) and Theorem 2.3 implies that

[TABLE]

Moreover, let $L_{\epsilon}$ and $\Theta^{\epsilon}$ is given in Theorem 4.1. The condition (2.5) implies that Lemma 4.8 (ii) holds, that is

[TABLE]

The proof of Theorem 2.4 is then divided into 3 steps.

Step 1. We first show that

[TABLE]

Here and below $f_{+}=\max\{f,0\}$. Observe that

[TABLE]

By Lemma 4.8 (i), we have

[TABLE]

By Lemma 4.8 (ii), we also have

[TABLE]

Step 2. We show that

[TABLE]

Taking $v=(u^{\epsilon}-x_{n})^{2}$ in Lemma 4.3, we have

[TABLE]

and hence, by (2.4),

[TABLE]

Since $\mathscr{A}_{H}(u^{\epsilon})+\epsilon\Delta u^{\epsilon}=0$, one has

[TABLE]

and hence, by (2.4),

[TABLE]

By Young’s inequality,

[TABLE]

By the strongly concavity/convexity of $H$, we know that

[TABLE]

and

[TABLE]

Thus

[TABLE]

Pluging this in (5.1), one gets

[TABLE]

Note that

[TABLE]

By Young’s inequality,

[TABLE]

Step 3. Set

[TABLE]

where $\zeta\in C^{\infty}_{c}(B(0,2))$, $0\leq\zeta\leq 1$ in $B(0,2)$ and $\zeta=1$ in $B(0,1)$. Lemma 4.3 gives that

[TABLE]

where we used $\zeta|_{\partial B(0,2)}=0$. One has

[TABLE]

Owing to $L_{\epsilon}(u^{\epsilon}_{s})=0$, we further compute

[TABLE]

Note that

[TABLE]

and hence

[TABLE]

and

[TABLE]

We conclude that

[TABLE]

In view of (5.2), we conclude that

[TABLE]

Since $H$ is strongly convex,

[TABLE]

This implies that

[TABLE]

The proof of Theorem 2.4 is complete.

Acknowledgment. The authors would like to thank the supports of National Natural Science of Foundation of China (No. 11522102&11871088).