A note on boundary differentiability of solutions of nondivergece elliptic equations with unbounded drift

TL;DR

This paper demonstrates boundary differentiability of solutions to nondivergence elliptic equations with unbounded drift, advancing understanding of solution regularity under challenging conditions.

Contribution

It establishes boundary differentiability results for elliptic equations with unbounded drift, a case less understood in existing literature.

Findings

Boundary differentiability is proven for solutions with unbounded drift.

The results extend regularity theory to more general elliptic equations.

The paper provides new techniques for handling unbounded coefficients.

Abstract

Boundary differentiability is shown for solutions of nondivergence elliptic equations with unbounded drift

Equations

$$\displaystyle\left\{\begin{array}[]{l}Lu:=-a_{ij}(x)D_{ij}u+b_{i}(x)D_{i}u=f(x)\;\;in\;\;\ Q^{+}_{1};\\ u(x)=0\;\;on\;\;\ T_{1}.\end{array}\right.$$

$$\lambda |\xi {|}^2\le a_{ij}(x){\xi }_i{\xi }_j\le {\lambda }^{-1}|\xi {|}^2,\forall \xi \in {\mathbb{R}}^n,a.e.x\in Q_1^{+},$$

$$\underset{\Omega }{\sup }u\le \underset{\partial \Omega }{\sup }u+Ndiam\Omega \cdot e^{N||\mathbf{b}|{|}_{L^n(\Omega )}^n}||f^{+}|{|}_{L^n(\Omega )},$$

$$||\mathbf{b}||_{L^{n}(\Omega)}=\Big{(}\int_{\Omega}|\mathbf{b}|^{n}dx\Big{)}^{\frac{1}{n}},\ \ \ \mathbf{b}=(b_{1},b_{2},...,b_{n}),$$

$$\underset{B_1}{\sup }u\le C(\underset{B_1}{\inf }u+||f|{|}_{L^n(B_8)}),$$

$$||\mathbf{b}|{|}_{L^n(Q_1^{+})}\le \min \{\frac{\delta }{4A_2(\frac{4M}{\mu }+1)},1,{ϵ}_0\}and{\int }_0^1\frac{||b_n|{|}_{L^n(Q_r^{+})}}{r}dr\le \min \{1,\frac{\delta \ln \frac{1}{\delta }}{16M{A}_3}\},$$

$$k{x}_n-B\le u(x)\le K{x}_n+B\text{mbox}in{Q}_1^{+},$$

$$\tilde{k}{x}_n-A_1||f|{|}_{L^n(Q_1^{+})}-A_2(K-k+B)||\mathbf{b}|{|}_{L^n(Q_1^{+})})-A_3(|K|+|k|)||b_n|{|}_{L^n(Q_1^{+})}$$

$$\le u(x)\le \tilde{K}{x}_n+A_1||f|{|}_{L^n(Q_1^{+})}+A_2(K-k+B)||\mathbf{b}|{|}_{L^n(Q_1^{+})})+A_3(|K|+|k|)||b_n|{|}_{L^n(Q_1^{+})},$$

$$\tilde{k}=k-2MB+\mu (K-k)\text{mbox}and\tilde{K}=K+2MB,$$

$$\tilde{k}=k-2MB\text{mbox}and\tilde{K}=K+2MB-\mu (K-k).$$

$$(k-2MB)x_n-C_1(|k|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})})$$

$$\le u(x)\le (K+2MB)x_n+C_1(|K|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})}).$$

$$4-(1+ϵ)(2+ϵ)(M-1{)}^{ϵ}\ge 0.$$

$$\tilde{\delta }=\frac{1}{M}(\le \frac{1}{3\sqrt{n-1}}),\delta =\frac{\tilde{\delta }}{2M}$$

$$\tilde{\psi}(x)=2\Big{(}\frac{x_{n}}{\tilde{\delta}}\Big{)}-\Big{(}\frac{x_{n}}{\tilde{\delta}}\Big{)}^{2}+\frac{\lambda^{2}}{2(n-1)}\sum_{i=1}^{n-1}\left(\left(\frac{|x_{i}|}{\tilde{\delta}}-1\right)^{+}\right)^{2+\epsilon}.$$

$$\displaystyle\left\{\begin{array}[]{l}(1)\tilde{\psi}(x)\geq 1\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|\leq 1,x_{n}=\tilde{\delta}\};\\[3.0pt] (2)\tilde{\psi}(x)\geq 0\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|\leq 1,x_{n}=0\};\\[3.0pt] (3)\tilde{\psi}(x)\geq 1\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|=1,0\leq x_{n}\leq\tilde{\delta}\};\\[3.0pt] (4)-a^{ij}(x)D_{ij}\tilde{\psi}(x)\geq 0\ \ \mbox{a.e.}\ \mbox{in}\ \ \{x\in\mathbb{R}^{n}:|x^{\prime}|<1,\ \ \ 0<x_{n}<\tilde{\delta}\};\\[3.0pt] (5)\tilde{\psi}(x)\leq\frac{2x_{n}}{\tilde{\delta}}\ \ \ \mbox{in}\ \ Q^{+}_{\tilde{\delta}}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{l}L\big{(}kx_{n}-B\tilde{\psi}(x)-u(x)\big{)}\leq b_{i}D_{i}(kx_{n}-B\tilde{\psi}(x))-f(x)\;\;in\;\;\ \tilde{Q};\\[10.0pt] kx_{n}-B\tilde{\psi}(x)-u(x)\leq 0\;\;on\;\;\partial\tilde{Q},\end{array}\right.$$

$$k{x}_n-B\tilde{\psi }(x)-u(x)\le C_1(|k|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})}))\text{mbox}in\tilde{Q},$$

$$(k-2MB)x_n-C_1(|k|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})})\le u(x)\text{mbox}in{Q}_{\tilde{\delta }}^{+}.$$

$$\displaystyle\left\{\begin{array}[]{l}L\big{(}u(x)-Kx_{n}-B\tilde{\psi}(x)\big{)}\leq-b_{i}D_{i}(Kx_{n}+B\tilde{\psi(x)})+f(x)\;\;in\;\;\ \tilde{Q};\\[10.0pt] u(x)-Kx_{n}-B\tilde{\psi}(x)\leq 0\;\;on\;\;\partial\tilde{Q}.\end{array}\right.$$

$$u(x)-K{x}_n-B\tilde{\psi }(x)\le C_1(|K|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})})\text{mbox}in\tilde{Q},$$

$$u(x)\le (K+2MB)x_n+C_1(|K|||b_n|{|}_{L^n(Q_1^{+})}+B||\mathbf{b}|{|}_{L^n(Q_1^{+})}+||f|{|}_{L^n(Q_1^{+})})\text{mbox}in{Q}_{\tilde{\delta }}.$$

$$v(\delta e_{n})\geq\big{(}\frac{K-k}{2}+2MB\big{)}\delta+C_{1}(|k|||b_{n}||_{L^{n}(Q^{+}_{1})}+B||\mathbf{b}||_{L^{n}(Q^{+}_{1})}+||f||_{L^{n}(Q^{+}_{1})}).$$

$$\sup_{\Gamma}v(x)\leq C_{2}\big{(}\inf_{\Gamma}v(x)+|k|||b_{n}||_{L^{n}(Q^{+}_{1})}+B||b_{n}||_{L^{n}(Q^{+}_{1})}+||f||_{L^{n}(Q^{+}_{1})}\big{)},$$

$$\inf_{\Gamma}v(x)\geq\Big{\{}\frac{1}{C_{2}}\Big{(}(\frac{K-k}{2}+2MB)\delta\Big{)}+(\frac{C_{1}}{C_{2}}-1)(|k|||b_{n}||_{L^{n}(Q^{+}_{1})}+B||\mathbf{b}||_{L^{n}(Q^{+}_{1})}+||f||_{L^{n}(Q^{+}_{1})}\Big{\}}^{+}:=a.$$

$$\displaystyle\psi(x)=\frac{1}{2}\left(\Big{(}\frac{x_{n}}{\delta}\Big{)}+\Big{(}\frac{x_{n}}{\delta}\Big{)}^{2}\right)-\frac{\lambda^{2}}{4(n-1)}\sum_{i=1}^{n-1}\left(\left(\frac{|x_{i}|}{\delta}-1\right)^{+}\right)^{2+\epsilon},$$

$$\displaystyle\left\{\begin{array}[]{l}(1)\psi(x)\leq 1\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|\leq M\delta,x_{n}=\delta\};\\[3.0pt] (2)\psi(x)\leq 0\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|\leq M\delta,x_{n}=0\};\\[3.0pt] (3)\psi(x)\leq 0\ \mbox{on}\ \{x\in\mathbb{R}^{n}:\ |x^{\prime}|=M\delta,0<x_{n}<\delta\};\\[3.0pt] (4)-a_{ij}(x)D_{ij}\psi(x)\leq 0\ \ \mbox{a.e.\ in}\ \ \{x\in\mathbb{R}^{n}:|x^{\prime}|<M\delta,\ 0<x_{n}<\delta\};\\[3.0pt] (5)\psi(x)\geq\frac{x_{n}}{2\delta}\ \ \ \mbox{in}\ \ Q^{+}_{\delta};\\[3.0pt] \end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{l}L\big{(}a\psi(x)-v(x)\big{)}\leq b_{i}D_{i}(a\psi(x)+(k-2MB)x_{n})-f(x)\;\;\;\;in\;\;\ \tilde{\tilde{Q}};\\[10.0pt] a\psi(x)-v(x)\leq 0\;\;on\;\;\partial\tilde{\tilde{Q}},\end{array}\right.$$

$$a\psi (x)-v(x)\le C_3(K-k+B)||\mathbf{b}|{|}_{L^n(Q_1^{+})}+C_4|k|||b_n|{|}_{L^n(Q_1^{+})}+C_5||f|{|}_{L^n(Q_1^{+})}\text{mbox}in\widetilde{\stackrel{~}{Q}},$$

$$a\psi (x)$$

Full text

††journal: www.arxiv.org

1 Introduction

In this paper, we will study the boundary differentiability of strong solutions of elliptic equation with unbounded lower order coefficients. Suppose that $u\in W^{2,n}_{loc}(Q^{+}_{1})\cap C(\overline{Q^{+}_{1}})$ satisfies

[TABLE]

We use the summation convention over repeated indices and the notations $D_{i}:=\frac{\partial}{\partial x_{i}}$; $D_{ij}:=D_{i}D_{j}$. We assume that $a_{ij}$, $b_{i}$ and $f$ are measurable functions on $Q^{+}_{1}$, $\mathbf{b}=(b_{1},b_{2},...,b_{n})$, the matrix $(a_{ij}(x))_{n\times n}$ is symmetric and satisfies the uniformly elliptic condition

[TABLE]

with a constant $\lambda\in(0,1]$, and $b_{i},f\in L^{n}(Q^{+}_{1})$. Throughout the paper, we denote $W(\Omega):=W^{2,n}_{loc}(\Omega)\cap C(\overline{\Omega})$ while $\Omega$ is a bounded domain in $\mathbb{R}^{n}$.

As for the boundary regularity of nondivergence elliptic equations: If the drift term $|\mathbf{b}|$ is bounded, Krylov [8] showed that the solution is $C^{1,\alpha}$ along the boundary if $\partial\Omega$ is $C^{1,1}$; Lieberman [9] gave a more general estimates; Wang [16] proved a similar pointwise result as in [8] by an iteration method that will be adopted in this paper; Li and Wang in [10, 11] showed the boundary differentiability of solutions of elliptic equations on convex domains. If $|\mathbf{b}|$ is unbounded, Ladyzhenskaya and Ural’tseva [5] proved boundary $C^{1,\alpha}$ estimate of elliptic and parabolic inequalities on $W^{2,q}$ domain with $\mathbf{b}\in L^{q}$, $\Phi\in L^{q}$, $q>n$ and nonlinear term $\mu_{1}|Du|^{2}$; Safonov [15] obtained the the Hopf-Oleinik lemma for elliptic equations and gave the counterexample which indicated that the Dini condition on $b_{n}$ can not be removed for our theorem; Nazarov [13] proved the Hopf-Oleinik Lemma and boundary gradient estimate under minimal restrictions on lower-order coefficients; In [12] the boundary differentiability is shown for strong solution of nondivergence elliptic equation $|\mathbf{b}|$ and $f$ satisfying Dini condition. Since the Hopf Oleinik Lemma and boundary Lipschitz Estimate [13] hold for solution of (1.1) only need $b_{n}$ satisfies the Dini condition, it is natural conjecture that whether the boundary differentiability of solutions at [math] is true while $b_{n}$ satisfying Dini condition at [math]. In the following, we will show that the result is correct. Some related results concerning Dini continuity can be found in [3, 6, 7, 14].

The following Alexandroff-Bakelman-Pucci maximum principle and Harnack inequality are our main tools.

Theorem 1.1**.**

([4, 15]) Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and let $u$ be a function in $W(\Omega)$ such that $Lu\leq f$ in $\Omega$. Suppose that the matrix $(a_{ij}(x))_{n\times n}$ is symmetric and satisfies the uniformly elliptic condition (1.4), and $b_{i},f\in L^{n}(\Omega)$. Then

[TABLE]

where

[TABLE]

and $N$ is a positive constant depending only on $n$ and $\lambda$.

Theorem 1.2**.**

(Harnack Inequality) Let $u$ be a nonnegative function in $W(B_{8})$, $Lu=f$ in $B_{8}$ and $b_{i},f\in L^{n}(B_{8})$. There exists a positive constant $\epsilon_{0}$ depending only on $\lambda$ and $n$, such that if $||\mathbf{b}||_{L^{n}(B_{8})}\leq\epsilon_{0}$, then

[TABLE]

where $C$ is constant depending only on $\lambda$ and $n$.

Theorem 1.2 follows from the the proof in [15] clearly. The most important thing is that the quantity $||\mathbf{b}||_{L^{n}}$ is scaling invariant(see Remark 1.4 in [15]) and the Harnack constant is invariant in the iteration procedure.

Notations.

$\{\vec{e}_{i}\}_{i=1}^{n},$                            the standard basis of $\mathbb{R}^{n}$.

$|x|~{}:=\sqrt{\sum\limits_{i=1}^{n}x_{i}^{2}},$           the Euclidean norm of $x=(x_{1},x_{2},...,x_{n})\in\mathbb{R}^{n}$.

$B_{r}:=\{x\in\mathbb{R}^{n}:\ |x|<r\}$.

$T_{r}:=\{x^{\prime}\in\mathbb{R}^{n-1}:\ |x^{\prime}|<r\}.$

$Q_{r}:=T_{r}\times(-r,r).$

$Q_{r}^{+}:=T_{r}\times(0,r).$

||f||_{L^{n}(\Omega)}:=\Big{(}\int_{\Omega}|f(x)|^{n}dx\Big{)}^{\frac{1}{n}}.

$W(\Omega):=W^{2,n}_{loc}(\Omega)\cap C(\overline{\Omega}).$

Theorem 1.3**.**

Assume that (1)$u\in W(Q^{+}_{1})$, $u|_{T_{1}}=0$, $Lu=f$ in $Q^{+}_{1}$; (2)$f\in L^{n}(Q^{+}_{1})$ and $\int_{0}^{1}\frac{||f||_{L^{n}(Q^{+}_{r})}}{r}dr<\infty$; (3)$\mathbf{b}\in L^{n}(Q^{+}_{1})$ and $\int_{0}^{1}\frac{||b_{n}||_{L^{n}(Q^{+}_{r})}}{r}dr<\infty$. Then $u$ is differentiable at [math].

2 Proof of Theorem

By standard normalization, it is enough for us to prove the following Theorem 2.1 instead of proving Theorem 1.3.

Theorem 2.1**.**

*Assume that

(1)$u\in W(Q^{+}_{1})$, $||u||_{L^{\infty}(Q^{+}_{1})}\leq 1$, $Lu=f$ in $Q^{+}_{1}$,  and    $u|_{T_{1}}=0$;

(2)$f\in L^{n}(Q^{+}_{1})$ with $||f||_{L^{n}(Q^{+}_{1})}\leq 1$ and $\int_{0}^{1}\frac{||f||_{L^{n}(Q^{+}_{r})}}{r}dr\leq 1$;

(3)$\mathbf{b}$ and $b_{n}$ satisfy*

[TABLE]

*where $\epsilon_{0}$ is the constant in Theorem 1.2, and $\delta$, $M$, $\mu$, $A_{2}$ and $A_{3}$ are constants in Lemma 2.2.

Then $u$ is differentiable at [math].*

Lemma 2.2**.**

There exist positive constant $\delta(<1)$,$\mu(<1)$, $M$, $A_{1}$, $A_{2}$ and $A_{3}$ depending only on $\lambda$ and $n$. If

[TABLE]

for some constants $k$, $K$ and $B(\geq 0)$ with $k\leq K$, then there exist constants $\tilde{k}$ and $\tilde{K}$ such that for $x\in Q^{+}_{\delta}$,

[TABLE]

where either

[TABLE]

or

[TABLE]

Obviously, we have $\tilde{k}\leq\tilde{K}$.

Proof of Lemma 2.2.

We prove the following claim first.

Claim. There exist positive constants $M$, $\tilde{\delta}$ and $C_{1}$ depending only on $\lambda$ and $n$, such that for any $x\in Q^{+}_{\tilde{\delta}}$,

[TABLE]

Proof.

Let $M\geq\sqrt{n-1}(1+\frac{2\sqrt{n-1}}{\lambda})$ and $\epsilon(>0)$ be small enough, such that

[TABLE]

Let

[TABLE]

and

[TABLE]

The barrier function $\tilde{\psi}(x)$ is $C^{2}$ and satisfies the following conditions:

[TABLE]

It follows that

[TABLE]

where $\tilde{Q}=T_{1}\times(0,\tilde{\delta}).$

According to the Alexandroff-Bakelman-Pucci maximum principle, we have

[TABLE]

where $C_{1}$ is a constant depending only on $\lambda$ and $n$.

By (2.10)(5)(i.e. $\tilde{\psi}(x)\leq\frac{2x_{n}}{\tilde{\delta}}=2Mx_{n}\ \ \mbox{in}\ Q^{+}_{\tilde{\delta}}$), we have

[TABLE]

As in (2.11), we also have

[TABLE]

According to the Alexandroff-Bakelman-Pucci maximum principle, we have

[TABLE]

where $C_{1}$ is a constant depending only on $\lambda$ and $n$ . Combining (2.15) and (2.10)(5), we get

[TABLE]

By (2.13) and (2.16), the claim follows clearly.∎

Let $\Gamma=\{\delta\vec{e}_{n}+T_{M\delta}\}$. Next, we will show (2.3) according to two cases: $u(\delta e_{n})\geq\frac{1}{2}(K+k)\delta$ and $u(\delta e_{n})<\frac{1}{2}(K+k)\delta$, corresponding to which (2.4) and (2.5) will hold respectively. Since the proofs of these two cases are similar, we will only show the proof for the case: $u(\delta e_{n})\geq\frac{1}{2}(K+k)\delta$.

Let $v(x)=u(x)-(k-2MB)x_{n}+C_{1}(|k|||b_{n}||_{L^{n}(Q^{+}_{1})}+B||\mathbf{b}||_{L^{n}(Q^{+}_{1})}+||f||_{L^{n}(Q^{+}_{1})})$. Then

[TABLE]

Since $v(x)\geq 0$ for $x\in Q^{+}_{\tilde{\delta}}$, from (2.17) and the Harnack inequality, it follows that

[TABLE]

where $C_{2}(\gg 1)$ is a constant depending only on $\lambda$ and $n$. Combining (2.17),(2.18) and $v(x)\geq 0$, we have

[TABLE]

Let

[TABLE]

where $\epsilon$ satisfies (2.7).

The barrier function $\psi(x)$ is $C^{2}$ and satisfies the following conditions :

[TABLE]

It follows that

[TABLE]

where $\tilde{\tilde{Q}}=T_{M\delta}\times(0,\delta).$

According to the Alexandroff-Bakelman-Pucci maximum principle,

[TABLE]

where $C_{3},C_{4},C_{5}$ are constants depending only on $\lambda$ and $n$, and we have used $K-k\geq 0$.

From (2.21)(5), it follows that for each $x\in Q^{+}_{\delta}$,

[TABLE]

Combining (2.23) and (2.24), we have that for each $x\in Q^{+}_{\delta}$,

[TABLE]

Let

[TABLE]

Combining (2.16),(2.25) and (2.26) , we have that (2.3) and (2.4) hold.∎

By induction, the following lemma is a direct consequence of Lemma 2.2.

Lemma 2.3**.**

There exist sequences $\{k_{m}\}_{m=0}^{\infty}$ and $\{K_{m}\}_{m=0}^{\infty}$, and nonnegative sequence $\{B_{m}\}_{m=0}^{\infty}$ with $k_{0}=K_{0}=0$ , $B_{0}=1$, and for $m=0,1,2,...,$

[TABLE]

and

[TABLE]

or

[TABLE]

such that

[TABLE]

where $\delta$, $\mu$, $M$, $A_{1}$ and $A_{2}$ are positive constants given by Lemma 2.2.

Now we present the proof of Theorem 2.1.

Proof of Theorem 2.1.

Let $\{B_{m}\}_{m=0}^{\infty}$, $\{k_{m}\}_{m=0}^{\infty}$ and $\{K_{m}\}_{m=0}^{\infty}$ be defined by Lemma 2.3. We will show the proof by the following three claims.

Claim 1. $\sum\limits_{m=0}^{\infty}\frac{B_{m}}{\delta^{m}}$ is convergent.

Proof.

Firstly, notice that we take $K_{0}=k_{0}=0$ and $B_{0}=1$, then by induction, we have $K_{m}\geq k_{m}$ for all $m\geq 0$.

For $m\geq 0$, we define $S_{m}=\sum\limits_{i=0}^{m}\frac{B_{i}}{\delta^{i}}.$ For any $m\geq 0$, since

[TABLE]

we have

[TABLE]

Similarly, we have

[TABLE]

Hence,

[TABLE]

Now we consider the term $K_{m}-k_{m}$. By Lemma 2.3, for any $m\geq 0$,

[TABLE]

Since $K_{0}=k_{0}=0$, by iteration, we have that for any $m\geq 0$,

[TABLE]

It follows that for $m\geq 1$,

[TABLE]

By changing the order of summation, we have

[TABLE]

By

[TABLE]

we have that for $m\geq 1$,

[TABLE]

Since

[TABLE]

for any $i\geq 1$, combining the above identity with (2.28), we obtain

[TABLE]

It follows from (2.30) and (2.31) that for any $m\geq 1$,

[TABLE]

Since

[TABLE]

and

[TABLE]

it follows that

[TABLE]

Therefore for all $m\geq 1$,

[TABLE]

where we used $||f||_{L^{n}(Q^{+}_{1})}\leq 1$, $\int_{0}^{1}\frac{||f||_{L^{n}(Q^{+}_{r})}}{r}dr\leq 1$ and $||\mathbf{b}||_{L^{n}(Q^{+}_{1})}\leq 1$. Then $\{S_{m}\}_{m=0}^{\infty}$ is a uniformly bounded sequence. It follows that $\sum\limits_{m=0}^{\infty}\frac{B_{m}}{\delta^{m}}$ is convergent. This completes the proof of Claim 1.∎

Claim 2.

[TABLE]

Proof.

It follows from Claim 1 that $\{K_{m}\}_{m=0}^{\infty}$ and $\{k_{m}\}_{m=0}^{\infty}$ are uniformly bounded.

Since

[TABLE]

we obtain

[TABLE]

It follows that $\{K_{m}-2MS_{m-1}\}_{m=1}^{\infty}$ is a bounded nonincreasing sequence and $\lim\limits_{m\rightarrow\infty}(K_{m}-2MS_{m-1})$ exists. Hence $\lim\limits_{m\rightarrow\infty}K_{m}$ exists. Let $\lim\limits_{m\rightarrow\infty}K_{m}=\theta$.

Since

[TABLE]

we have $\sum\limits_{j=0}^{\infty}(K_{j}-k_{j})$ is convergent. It follows that $\lim\limits_{m\rightarrow\infty}(K_{m}-k_{m})=0$. Hence

[TABLE]

This completes the proof of Claim 2.∎

Claim 3. Let $\theta$ be given by Claim 2. Then for each $m=0,1,2,...$, there exist $C_{m}$ such that $\lim\limits_{m\rightarrow\infty}C_{m}=0$ and that $|u(x)-\theta x_{n}|\leq C_{m}\delta^{m}$ for any $x\in Q^{+}_{\delta^{m}}$.

Proof.

For any $m\geq 0$ and any $x\in Q^{+}_{\delta^{m}}$, we have

[TABLE]

Let $C_{m}=|K_{m}-\theta|+|k_{m}-\theta|+\frac{B_{m}}{\delta^{m}}$. It follows that for any $m\geq 0$ and any $x\in Q^{+}_{\delta^{m}}$,

[TABLE]

and

[TABLE]

This completes the proof of Claim 3.∎

By Claim 3, we deduce that $u(x)$ is differentiable at [math] with derivative $\theta\vec{e}_{n}$. This completes the proof of Theorem 2.1. ∎