Global weighted gradient estimates for nonlinear p-Laplacian type elliptic equations and its application

TL;DR

This paper establishes global weighted gradient estimates for solutions to nonlinear p-Laplacian type elliptic equations on Reifenberg flat domains, and explores Besov regularity for certain harmonic equations, advancing understanding of regularity in complex geometries.

Contribution

It provides new global weighted $W^{1,p}$ estimates for nonlinear elliptic equations with specific regularity assumptions on the coefficients, and derives Besov regularity results for special harmonic equations.

Findings

Established global weighted $W^{1,p}$ estimates for nonlinear elliptic equations.

Proved Besov regularity for solutions of certain harmonic equations.

Extended regularity theory to Reifenberg flat domains.

Abstract

We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,\xi)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm in $x$. Moreover, we derive Besov regularity for solutions of a class of special harmonic equations by making use of $W^{1,p}$ estimate. Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity

Equations

$$\left\{\begin{array}[]{r@{\ }c@{\ }ll}\operatorname{div}A(x,u,\nabla u)&=&\operatorname{div}\left(|F|^{p-2}F\right)&\mbox{in}\ \ \Omega\,,\\[1.42271pt] u&=&0&\mbox{on}\ \ \partial\Omega.\\[1.42271pt] \end{array}\right.$$

$$|A(x,z_1,\xi )-A(x,z_2,\xi )|\le {\omega }_M(|z_1-z_2|)|\xi {|}^{p-1}$$

$$\left\{\begin{array}[]{c}|A(x,z,\xi)|+|\partial_{\xi}A(x,z,\xi)||\xi|\leq\Lambda|\xi|^{p-1}\\ \left<\partial_{\xi}A(x,z,\xi)\zeta,\zeta\right>\geq\Lambda^{-1}|\xi|^{p-2}|\zeta|^{2}.\\ \end{array}\right.$$

$${\int }_{\Omega }⟨A(x,u,\nabla u),\nabla \phi ⟩dx={\int }_{\Omega }⟨|F{|}^{p-2}F,\nabla \phi ⟩dx$$

$$\theta \left(A,B_{\rho }(y)\right)(x,z)=\underset{\xi \in {\mathbb{R}}^n\setminus \{0\}}{\sup }\frac{|A(x,z,\xi )-{\bar{A}}_{B_{\rho }(y)}(z,\xi )|}{|\xi {|}^{p-1}}$$

$${\bar{A}}_{B_{\rho }(y)}(z,\xi )={\text{fint}}_{B_{\rho }(y)}A(x,z,\xi )dx$$

$$B_{\rho }(x_0)\cap \{x:x_n>\rho \delta \}\subset B_{\rho }(x_0)\cap \Omega \subset B_{\rho }(x_0)\cap \{x:x_n>-\rho \delta \}$$

$$[\omega {]}_q:=\underset{ballsB\subset {\mathbb{R}}^n}{\sup }\left({\text{fint}}_B\omega (x)dx\right){\left({\text{fint}}_B\omega (x{)}^{\frac{1}{1-q}}dx\right)}^{q-1}<+\infty .$$

$$\Vert g{\Vert }_{L_{\omega }^{q,t}(\Omega )}:={\left(q{\int }_0^{\infty }{\left({\alpha }^q\omega (\{x\in \Omega :|g(x)|>\alpha \})\right)}^{\frac{t}{q}}\frac{d\alpha }{\alpha }\right)}^{\frac{1}{t}}<+\infty$$

$$\Vert g{\Vert }_{L_{\omega }^{q,\infty }(\Omega )}:=\underset{\alpha >0}{\sup }\alpha \omega (\{x\in \Omega :|g(x)|>\alpha \}{)}^{\frac{1}{q}}.$$

$$\Vert g{\Vert }_{L_{\omega }^q(\Omega )}:={\left({\int }_{\Omega }|g(x){|}^q\omega (x)dx\right)}^{\frac{1}{q}}$$

$$\underset{-M\le z\le M}{\sup }\underset{0<\rho \le R}{\sup }\underset{y\in {\mathbb{R}}^n}{\sup }{\text{fint}}_{B_{\rho }(y)}\theta \left(A,B_{\rho }(y)\right)(x,z)dx\le \delta$$

$$\Vert \nabla u{\Vert }_{L_{\omega }^{pq,t}(\Omega )}\le C\Vert F{\Vert }_{L_{\omega }^{pq,t}(\Omega )}$$

$$\mathrm{div}A(x,u,\nabla u)=\mathrm{div}\left(|F{|}^{p-2}F\right)\text{mbox}in{B}_{2R}$$

$$\underset{-M\le z\le M}{\sup }\underset{0<\rho \le R}{\sup }\underset{y\in B_R}{\sup }{\text{fint}}_{B_{\rho }(y)}\theta \left(A,B_{\rho }(y)\right)(x,z)dx\le \delta$$

$$\Vert \nabla u{\Vert }_{L_{\omega }^{pq,t}(B_R)}\le C\left(\Vert F{\Vert }_{L_{\omega }^{pq,t}(B_{2R})}+\omega (B_{2R}{)}^{1/pq}{\left({\text{fint}}_{B_{2R}}|\nabla u{|}^{p}dx\right)}^{1/p}\right)$$

$$\Vert f{\Vert }_{B_{p,q}^{\alpha }({\mathbb{R}}^n)}=\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}+[f{]}_{{\dot{B}}_{p,q}^{\alpha }({\mathbb{R}}^n)}$$

$$[f{]}_{{\dot{B}}_{p,q}^{\alpha }({\mathbb{R}}^n)}={\left({\int }_{{\mathbb{R}}^n}{\left({\int }_{{\mathbb{R}}^n}\frac{|f(x+h)-f(x){|}^p}{|h{|}^{\alpha p}}dx\right)}^{\frac{q}{p}}\frac{dh}{|h{|}^n}\right)}^{\frac{1}{q}}.$$

$$\Vert f{\Vert }_{B_{p,\infty }^{\alpha }({\mathbb{R}}^n)}=\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}+[f{]}_{{\dot{B}}_{p,\infty }^{\alpha }({\mathbb{R}}^n)}$$

$$[f{]}_{{\dot{B}}_{p,\infty }^{\alpha }({\mathbb{R}}^n)}=\underset{h\in {\mathbb{R}}^n}{\sup }{\left({\int }_{{\mathbb{R}}^n}\frac{|f(x+h)-f(x){|}^p}{|h{|}^{\alpha p}}dx\right)}^{\frac{1}{p}}.$$

$$|A(x,z,\xi )-A(y,z,\xi )|\le |x-y{|}^{\alpha }(g(x)+g(y))|\xi |$$

$$\mathrm{div}A(x,u,\nabla u)=0\text{mbox}in\Omega ,$$

$$\hat{A}(x,z,\xi )=\frac{A(rx+x_0,\mu rz,\mu \xi )}{{\mu }^{p-1}}$$

$$\underset{-\frac{M}{\mu r}\le z\le \frac{M}{\mu r}}{\sup }\underset{0<\rho \le \frac{R}{r}}{\sup }\underset{y\in {\mathbb{R}}^n}{\sup }{\text{fint}}_{B_{\rho }(y)}\theta \left(A,B_{\rho }(y)\right)(x)dx\le \delta$$

$$|\hat{A}(x,z_1,\xi )-\hat{A}(x,z_2,\xi )|\le {\omega }_M(\mu r|z_1-z_2|)|\xi {|}^{p-1}$$

$$\left\{\begin{array}[]{r@{\ }c@{\ }ll}\operatorname{div}\hat{A}(x,\hat{u},\nabla\hat{u})&=&\operatorname{div}\left(|\hat{F}|^{p-2}\hat{F}\right)&\mbox{in}\ \ \Omega\,,\\[1.42271pt] \hat{u}&=&0&\mbox{on}\ \ \partial\Omega.\\[1.42271pt] \end{array}\right.$$

$$\omega (B)\le [\omega {]}_q{\left(\frac{|B|}{|E|}\right)}^q\omega (E)$$

$$\omega (E)\le C{\left(\frac{|E|}{|B|}\right)}^{\alpha }\omega (B)$$

$$g\in L_{\omega }^{q,t}(U)\iff S:=\sum _{k\ge 1}{\Gamma }^{tk}\omega {\left(\{x\in U:g(x)>\theta {\Gamma }^k\}\right)}^{\frac{t}{q}}<+\infty$$

$$C^{-1}S\le \Vert g{\Vert }_{L_{\omega }^{q,t}(U)}^t\le C\left(\omega (U{)}^{\frac{t}{q}}+S\right)$$

Full text

Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity

1 Introduction and main results.

1.1 Introduction.

In this paper we consider the following nonlinear elliptic equations:

[TABLE]

where $p\in(1,\infty)$, $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$ is a bounded and generally irregular domain. $F$ is a given measurable vector field function. The solution $u:\Omega\longrightarrow\mathbb{R}$ is a real-valued unknown function. The nonlinearity $A=A(x,z,\xi):\Omega\times R\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is differentiable with respect to $\xi\neq 0$. Moreover, $A(x,z,\xi)$ is assumed to have local uniform continuity in $z$, i.e.

[TABLE]

for almost every $x\in\Omega$, all $z_{1},z_{2}\in[-M,M]$. Where $\omega_{M}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is modulus of continuity with $\lim\limits_{\rho\rightarrow 0^{+}}\omega_{M}(\rho)=0$, monotonically non-decreasing and concave. And we further assume that there exists a constant $\Lambda>0$ such that

[TABLE]

for almost every $x\in\Omega$, all $z\in\mathbb{R}$ and all $\xi,\zeta\in\mathbb{R}^{n}\backslash\{0\}$. Furthermore, we require some more regularity on nonlinearity, namely we assume $A(x,z,\xi)$ is measurable in $\Omega$ for every $(z,\xi)\in\mathbb{R}\times\mathbb{R}^{n}\setminus\{0\}$ and has a sufficiently small $BMO$ (bounded mean oscillation) semi-norm in $x$. More precise description of these structural requirements will be given in the next subsection. As usual, we consider a function $u\in W_{0}^{1,p}(\Omega)$, which is a weak solution of (1.1) with $F\in L^{p}(\Omega,\mathbb{R}^{n})$, if

[TABLE]

for any test function $\varphi\in W_{0}^{1,p}(\Omega)$.

As a classical topic in the regularity theory of solutions to partial differential equations and systems, Calderón-Zygmund theory has been the theme of a number of contributions with different peculiarities. This theory traces its origins back to works of Calderón and Zygmund [5] in 1950s. They proved the $L^{p}$-estimate for the gradient of solutions to linear elliptic equations in the whole $\mathbb{R}^{n}$ by establishing the standard Calderón-Zygmund theory of singular integrals. As for the case of parabolic equations, that’s Fabes’s contribution [8]. For the nonlinear Calderón-Zymund theory, Iwaniec [10] first derived the Calderón-Zymund estimates for the $p$-Laplace equations via the sharp maximal operators and priori regularity estimates. As for weighted case, Mengesha and Phuc obtained the global regularity estimates in weighted Lorentz spaces, see [14].Caffarelli and Peral [4] obtained the $W^{1,p}$ regularity of solutions to fully nonlinear elliptic equations. In the case when $A=A(x,\nabla u)$, the results has been obtained by many researchers, see [3] for classical Lebesgue spaces and [2] for weighted Lebesgue spaces. As for the case $A(x,u,\nabla u)$, the authors succeeded to obtain interior gradient estimates when $u$ is bounded, see [16]. In the recent paper [1], the authors obtained global gradient estimates of (1.1) for classical Lebesgue spaces in the case when $u\in L^{\infty}(\Omega)$.

As for Besov regularity, see [6][12], in which the case that $A$ is independent on $z$ and corresponding obstacle problems have been studied. In the process, Calderón-Zygmund estimate play a crucial role.

The present article is a natural outgrowth of [1] and deals with global weighted $W^{1,p}$ theory for (1.1). In particular, we derive an extended version of the global $W^{1,p}$ estimate in the settings of the weighted Lorentz space. At the end of the paper, we derive Besov regularity for solutions of a class of special harmonic equations by making use of Calderón-Zygmund estimate.

This paper is organized as follows. In the next subsection, we give some notations and precise statement of the main results. In Section2, we state some elementary estimates which will be used frequently in the paper. In Section3 we present weighted good-$\lambda$ type inequality that will be essential for the proof of the main theorem. In Section4, the desired global weighted estimate is obtain. The last section contains the proof of Besov regularity for solutions.

1.2 Notations and main results.

Let us start by introducing a few notations to be used in what follows.

Throughout the paper, we denote by $|U|$ the integral $\int_{U}\operatorname{d}\!x$ for every measurable set $U\subset\mathbb{R}^{n}$. For an open set $\Omega\subset\mathbb{R}^{n}$, $\Omega_{\rho}(x)\triangleq\Omega\cap B_{\rho}(x)$, where $B_{\rho}(x)$ is a n-dimensional open ball. For the sake of convenience and simplicity, we employ the letter $C>0$ to denote any constants which can be explicitly computed in terms of known quantities such as $n,p,q$. Thus the exact value denoted by $C$ may change from line to line in a given computation.

To measure the oscillation of $A(x,z,\xi)$ in $x$-variables on $B_{\rho}(y)$, we consider a function defined by

[TABLE]

where

[TABLE]

In order to state our main results, we introduce the following definitions.

Definition 1.1**.**

The domain is said to be $(\delta,R)$-Reifenberg flat if there exist postive constants $\delta$ and $R$ with the property that for each $x_{0}\in\partial\Omega$ and each $\rho\in(0,R)$, there exist a local coordinate system $\{x_{1},\cdots,x_{n}\}$ with origin at the point $x_{0}$ such that

[TABLE]

Definition 1.2**.**

Let $1<q<\infty$, a non-negative, locally integrable function $\omega:\mathbb{R}\rightarrow[0,\infty)$ is said to be in the class $A_{q}$ of Muckenhoupt weight if

[TABLE]

Definition 1.3**.**

The weighted Lorentz space $L_{\omega}^{q,t}(\Omega)$ with $0<q<\infty$, $0<t\leq\infty$, is the set of measurable functions $g$ on $\Omega$ such that

[TABLE]

when $t\neq\infty$; for $t=\infty$ the space $L_{\omega}^{q,\infty}(\Omega)$ is set to be the usual Marcinkiewica space with quasinorm

[TABLE]

Remark 1.4*.*

When $t=q$, the Lorentz space $L_{\omega}^{q,q}(\Omega)$ is equivalent to weighted Lebesgue space $L_{\omega}^{q}(\Omega)$, whose norm is defined by

[TABLE]

The main result of this paper is the following global regularity estimates for weak solutions of (1.1) in weighted Lorentz space.

Theorem 1.5**.**

Let $p,q,\gamma\geq 1$. Then, there exists a sufficiently small constant $\delta=\delta(p,q,n,\Lambda,\gamma,M,\omega_{M})>0$ such that the following statement holds true. For a given vector field $F\in L_{\omega}^{pq,t}(\Omega,\mathbb{R}^{n})$, $0<t\leq\infty$, if $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ satisfying $\|u\|_{L^{\infty}(\Omega)}\leq M$ is a weak solution of (1.1) with $A(x,z,\xi)$ satisfying (1.2), (1.3) and

[TABLE]

for some $R>0$. $\Omega$ is $(\delta,R)$-Reifenberg flat. Then the following weighted regularity estimate holds.

[TABLE]

where $\omega\in A_{q}$ with $[\omega]_{q}\leq\gamma$, $\theta\left(A,B_{\rho}(y)\right)$ is defined in (1.4) and $C$ is a constant depending on $n$, $p$, $q$, $\Lambda$, $\gamma$, $M$, $\omega_{M}$, $\Omega$.

As for the interior case, the proof is similar to that of global case. Thus, we only state the result.

Theorem 1.6**.**

Let $p,q,\gamma\geq 1$. Then, there exists a sufficiently small constant $\delta=\delta(p,q,n,\Lambda,\gamma,M,\omega_{M})>0$ such that the following statement holds true. For a given vector field $F\in L_{\omega}^{pq,t}(B_{2R},\mathbb{R}^{n})$, $0<t\leq\infty$, if $u\in W_{loc}^{1,p}(B_{2R})\cap L^{\infty}(B_{2R})$ satisfying $\|u\|_{L^{\infty}(B_{2R})}\leq M$ is a weak solution of

[TABLE]

with $A(x,z,\xi)$ satisfying (1.2), (1.3) and

[TABLE]

for some $R>0$. Then the following weighted regularity estimate holds.

[TABLE]

where $\omega\in A_{q}$ with $[\omega]_{q}\leq\gamma$, $\theta\left(A,B_{\rho}(y)\right)$ is defined in (1.4) and $C$ is a constant depending on $n$, $p$, $q$, $\Lambda$, $\gamma$, $M$, $\omega_{M}$, $R$.

In order to state the other main result, which is actually a consequence of Theorem1.6, we recall the Besov space $B_{p,q}^{\alpha}(\mathbb{R}^{n})$.

Definition 1.7**.**

Let $h\in\mathbb{R}^{n}$, $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$. Let $0<\alpha<1$ and $1\leq p,q<\infty$. The Besov space consists of all functions $f\in L^{p}(\mathbb{R}^{n})$ for which the norm

[TABLE]

is finite. Where

[TABLE]

When $q=\infty$, we say that $f\in B_{p,\infty}^{\alpha}$, if

[TABLE]

is finite. Where

[TABLE]

Remark 1.8*.*

As matter of fact, one can simply integrates for $h\in B_{\delta}$ for a fixed $\delta>0$ when $q<\infty$ and take the supremum over $|h|\leq\delta$ to obtain an equivalent norm.

Theorem 1.9**.**

Let $0<\alpha<1$, Assume that $A(x,z,\xi)$ satisfies (1.2) and (1.3) for $p=2$, take $\omega_{M}(t)=t^{\alpha}$. Moreover, we suppose that there exists $g\in L_{loc}^{\frac{n}{\alpha}}(\Omega)$ such that

[TABLE]

for a.e.$x\in\Omega$, $\forall(z,\xi)\in\mathbb{R}\times\mathbb{R}^{n}$. If $u\in W_{loc}^{1,2}(\Omega)\cap L^{\infty}(\Omega)$ is a weak solution of

[TABLE]

then, $\nabla u\in B_{2,\infty}^{\alpha}$, locally.

2 Preliminaries.

2.1 Invariance.

We note that our equation is scaling invariant. Indeed, if $A(x,u,\nabla u)$ satisfies the conditions (1.2), (1.3) and (1.6), then for some fixed $\mu,r>0$, $x_{0}\in\mathbb{R}$, the rescaled nonlinearity

[TABLE]

satisfies (1.3). Moreover, $\hat{A}(x,z,\xi)$ satisfies

[TABLE]

and

[TABLE]

for a.e. $x\in\widehat{\Omega}$, $\forall z_{1},z_{2}\in\left[-\frac{M}{\mu r},\frac{M}{\mu r}\right]$. Where $\widehat{\Omega}=\left\{\frac{x-x_{0}}{r},x\in\Omega\right\}$ is $\left(\delta,\frac{R}{r}\right)$-Reifenberg flat.

The properties mentioned above are obvious owing to some elementary calculation. Let us now consider the invariance of equation (1.1) with respect to scaling. Assume that $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ is a weak solution of (1.1), then $\hat{u}=u(rx+x_{0})/\mu\in W_{0}^{1,p}(\widehat{\Omega})\cap L^{\infty}(\widehat{\Omega})$ satisfying $\|\hat{u}\|_{L^{\infty}(\widehat{\Omega})}\leq\frac{M}{\mu r}$ solve the equation

[TABLE]

where $\hat{F}(x)=\frac{F(rx+x_{0})}{\mu}$.

2.2 Muckenhoupt weights and weighted inequalities.

We will use the strong doubling property of $A_{q}$ weight stated below. Hereafter we denote by $\omega(\Omega)$ the integral $\int_{\Omega}\omega(x)\operatorname{d}\!x$

Lemma 2.1**.**

(cf.[7]). For $1<q<\infty$, the following statements hold true

1. (1)

if $\omega\in A_{q}$, then for every ball $B\subset\mathbb{R}^{n}$ and every measurable set $E\subset B$,

[TABLE] 2. (2)

if $\omega\in A_{q}$ with $[\omega]_{q}\leq\gamma$ for some given $\gamma\geq 1$, then there is $C=C(\gamma,n)$ and $\alpha=\alpha(\gamma,n)>0$ such that

[TABLE]

for every ball $B\subset\mathbb{R}^{n}$ and every measurable set $E\subset B$.

Lemma 2.2**.**

(cf.[9]). Let $\omega$ be an $A_{q}$ weight for some $1<q<\infty$. Then there exists $\sigma=\sigma(n,q,[\omega]_{q})>0$ such that $q-\sigma>1$ and $\omega\in A_{q-\sigma}$ with $[\omega]_{q-\sigma}\leq C(n,q,[\omega]_{q})$.

secondly, we state the following result which comes from standard measure theory.

Lemma 2.3**.**

Assume that $g\geq 0$ is a measurable function in a bounded subset $U\subset\mathbb{R}^{n}$. Let $\theta>0$, $\Gamma>1$ be constants, and let $\omega$ be a weight in $\mathbb{R}^{n}$. Then for $0<q,t<\infty$, we have

[TABLE]

and moreover, there exist a constant $C>0$ depending only on $\theta,\Gamma,t$, such that

[TABLE]

Analogously, for $0<q<\infty$ and $t=\infty$ we have

[TABLE]

Where $T$ is the quantity

[TABLE]

The following is a summary of embedding theorems that will be used later, see [9].

Lemma 2.4**.**

Let $\Omega$ be a bounded measurable subset of $\mathbb{R}^{n}$ and $\omega$ be an $A_{q}$ weight for $1<q<\infty$.

1. (1)

If $0<t\leq p_{1}<p_{2}\leq\infty$, then $L_{\omega}^{p_{2},\infty}(\Omega)\subset L_{\omega}^{p_{1},t}(\Omega)$. Moreover

[TABLE] 2. (2)

If $0<t\leq\infty$, $0<q<\infty$, then $L_{\omega}^{q,t}(\Omega)\subset L_{\omega}^{q,\infty}(\Omega)$.

Thirdly, we concern on the connection between the boundedness of the Hardy-Littlewood maximal operator on weighted spaces and the characterization of $A_{q}$ weight, which is crucial in treating our problem. For a given locally integrable function $f\in L_{\operatorname{loc}}^{1}(\mathbb{R}^{n})$, the Hardy-Littlewood maximal function is defined as

[TABLE]

For a function $f$ that is defined only on a bounded domain $U$, we define

[TABLE]

Where $\chi U$ is the characteristic function of the set $U$. The following boundedness of Hardy-Littlewood maximal operator $\mathcal{M}:L_{\omega}^{q,t}(\mathbb{R}^{n})\rightarrow L_{\omega}^{q,t}(\mathbb{R}^{n})$ is classical.

Lemma 2.5**.**

(cf.[14][15]). Let $\omega$ be an $A_{q}$ weight for some $1<q<\infty$. For any $0<t\leq\infty$, there exists a constant $C=C(n,q,t,[\omega]_{q})$ such that

[TABLE]

for all $f\in L_{\omega}^{q,t}(\mathbb{R}^{n})$. Conversely, if (2.4) holds for all $f\in L_{\omega}^{q,t}(\mathbb{R}^{n})$, then $\omega$ must be an $A_{q}$ weight.

Finally, we recall the following technical lemma, which will be used in the proof of the weighted estimates, which is originally due to [11][17]. The version given below is proved in [13]

Lemma 2.6**.**

Let $\Omega$ be a $(\delta,R)$-Reifenberg flat domain with $\delta<\frac{1}{4}$, Suppose $\omega\in A_{q}$ with $[\omega]_{q}\leq\gamma$ for some $1<q<\infty$ and some $\gamma\geq 1$. Suppose also that $C,D$ are measurable sets satisfying $C\subset D\subset\Omega$ and there are $\rho_{0}\in\left(0,\frac{R}{2000}\right)$ such that the sequence of balls $\{B_{\rho_{0}}(y_{i})\}_{i=1}^{L}$ with centers $y_{i}\in\overline{\Omega}$ covers $\Omega$, Assume that $\epsilon\in(0,1)$ such that the followings hold,

1. (1)

$\omega(C)<\epsilon\,\omega\left(B_{\rho_{0}}(y_{i})\right)$* for all $i=1,\cdots L$,* 2. (2)

for all $x\in\Omega$ and $\rho\in(0,2\rho_{0})$, if $\omega(C\cap B_{\rho}(x))\geq\epsilon\,\omega(B_{\rho}(x))$, then $B_{\rho}(x)\cap\Omega\subset D$.

Then

[TABLE]

2.3 A known approximation estimate.

For the sake of convenience and simplicity, we use the notation $u,F,A$ and $\Omega$ instead of $\hat{u},\hat{F},\hat{A}$ and $\widehat{\Omega}$ respectively. Let $\sigma\geq 6$ be a universal constant, let $u$ be a weak solution of

[TABLE]

We consider the limiting problem

• •

interior case:

[TABLE]

• •

boundary case

[TABLE]

for the interior case, $\bar{A}(\xi)$ is given by

[TABLE]

for the boundary case, $\bar{A}(\xi)$ is given by

[TABLE]

where

[TABLE]

We recall a known approximation estimate established in [1]. This approximation estimate will be used in the proof of Theorem1.5.

Lemma 2.7**.**

(interior case) For some fixed $\epsilon\in(0,1)$, there exists a constants $\sigma=\sigma(n,p,\Lambda,\omega_{M},M,\epsilon)\geq 6$ such that $u\in W_{0}^{1,p}(B_{\sigma})$ is a weak solution of $(2.5)$ with $\|u\|_{L^{\infty}(B_{\sigma})}\leq\frac{M}{\mu r}$ and satisfies

[TABLE]

Suppose also that there exists some positive number $\delta=\delta(\Lambda,\omega_{M},n,p,M,\epsilon)\in(0,\frac{1}{8})$ such that

[TABLE]

and

[TABLE]

Then there exists a weak solution $h\in W^{1,p}(B_{4})$of $(2.6)$ such that the following inequality holds

[TABLE]

Where $C=C(n,p,\Lambda)>1$.

Lemma 2.8**.**

(boundary case) For some fixed $\epsilon\in(0,1)$, there exists a constants $\sigma=\sigma(\Lambda,\omega_{M},n,p,M,\epsilon)\geq 6$ such that $u\in W_{0}^{1,p}(\Omega_{\sigma})$ is a weak solution of $(2.5)$ with $\|u\|_{L^{\infty}(\Omega_{\sigma})}\leq\frac{M}{\mu r}$ and satisfies

[TABLE]

Suppose also that there exists some positive number $\delta=\delta(\Lambda,\omega_{M},n,p,M,\epsilon)\in(0,\frac{1}{8})$ such that

[TABLE]

[TABLE]

and

[TABLE]

Then there exists a weak solution $h\in W^{1,p}(B_{4}^{+})$of $(2.7)$ such that the following inequality holds

[TABLE]

Where $\bar{h}$ is the zero extension of $h$ from $B_{4}^{+}$ to $B_{4}$, $C=C(\Lambda,n,p)>1$.

3 Weighted estimates.

Lemma 3.1**.**

Let $p\geq 1$, $\gamma>1$ and $\epsilon>0$ sufficiently small. Then there exists sufficiently large number $N=N(n,p,\Lambda)>1$, some positive number $\delta=\delta(n,p,\Lambda,\epsilon,\gamma,M,\omega_{M})>0$ and $\sigma=\sigma(n,p,\Lambda,\epsilon,M,\omega_{M})\geq 6$ such that the following statement holds. Suppose that $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (1.1) with $\|u\|_{L^{\infty}(\Omega)}\leq M$ and the nonlinearity $A(x,z,\xi)$ satisfies (1.6). If $\Omega$ is a $(\delta,R)$-Reifenberg flat domain and for $\forall y\in\Omega$, $\forall r\in\left(0,\frac{R}{\sigma}\right]$, we have

[TABLE]

then

[TABLE]

for $\omega\in A_{q}$ with $[\omega]_{q}\leq\gamma$ and $q>1$.

Proof.

We divide the proof into two steps.

$Step1$. We begin by proof an unweighted estimate.

Suppose that $\hat{u}\in W_{0}^{1,p}(\widehat{\Omega})$ is a weak solution of (2.5) with $\|\hat{u}\|_{L^{\infty}(\widehat{\Omega})}\leq\frac{M}{\mu r}$ and the nonlinearity $\hat{A}(x,z,\xi)$ satisfies

[TABLE]

If $\widehat{\Omega}$ is a $(\delta,\sigma)$-Reifenberg flat domain and

[TABLE]

then, we claim that

[TABLE]

In fact, For a given $\epsilon>0$, let $\epsilon^{\prime}=\epsilon^{\prime}(n,p,\Lambda,\epsilon)>0$ be a positive number to be determined later. Then, let $\delta=\delta(n,p,\Lambda,\epsilon^{\prime},M,\omega_{M})>0$ be the number defined in Lemma2.7 and Lemma2.8. We prove the claim (3.3) with this choice of $\delta$. By the assumption (3.2), we can discover that there exists $x_{0}$ such that

[TABLE]

Since $x_{0}\in B_{1}$, we can easily obtain $B_{\rho}\subset B_{\rho+1}(x_{0})$. For $\forall\rho\geq 6$, it follows that

[TABLE]

[TABLE]

Owing to the nonlinearity $\hat{A}(x,z,\xi)$ satisfies (3.1), all conditions in Lemma2.7 and Lemma2.8 are satisfied. Thus, one can find $H\in L^{\infty}(\widehat{\Omega}_{3})$ such that

[TABLE]

Take $N^{p}=\max\{4^{p}\left(\frac{7}{6}\right)^{n}C_{*}^{p},2^{n}\}$, we claim that

[TABLE]

In order to prove this statement, assume that $x$ is a point in the set on the left side of (3.6), for any $r^{\prime}>0$, if $r^{\prime}<2$, note that $B_{r^{\prime}}(x)\subset B_{3}$, as a result, we have

[TABLE]

If $r^{\prime}\geq 2$, then $B_{r^{\prime}}(x)\subset B_{2r^{\prime}}(x_{0})$, we have from this and (3.4) that

[TABLE]

Hence, we have proved that (3.6) holds. It follows that

[TABLE]

In addition, owing to the weak (1,1)-type estimate of Hardy-Littlewood maximal function, we have

[TABLE]

Then we can get

[TABLE]

where the last inequality is due to (3.5). Finally, the estimate of (3.3) follows by making use of the definition of $E$ and choosing $\epsilon^{\prime}=\epsilon^{\prime}(n,p,\Lambda,\epsilon)$ such that $C^{\prime}(n,p,\Lambda,\gamma)\epsilon^{\prime p}=\epsilon$

$Step2.$ We will use properties of $A_{q}$ weights and the translation scaling invariance of Lebesgue measure to obtain a weighted version.

For $\forall y\in\Omega$, define

[TABLE]

[TABLE]

then, $\hat{A}(x,z,\xi)$ satisfies (3.1), $\hat{u}\in W_{0}^{1,p}(\widehat{\Omega})$ is weak solution of (2.5) with $\|\hat{u}\|_{L^{\infty}(\widehat{\Omega})}\leq\frac{M}{\mu r}$ and $\widehat{\Omega}$ is $(\delta,\frac{R}{r})$-Reifenberg flat domain. By the assumption, there exists $x_{0}\in\Omega_{\rho}(y)$ such that

[TABLE]

and

[TABLE]

then we can derive that $z_{0}=\frac{x_{0}-y}{r}\in B_{1}$ and $z_{0}\in\widehat{\Omega}$, it follows that

[TABLE]

Similarily,

[TABLE]

Hence, all conditions in $Step1$ are satisfied and as can be seen from the above process

[TABLE]

From $Step1$, we have

[TABLE]

Since Lebesgue measure is scale and translation invariant, it follows that

[TABLE]

where we used (3.8). Combining this and Lemma2.1(2), we can derive that

[TABLE]

Thus, the Lemma follows in view of the arbitrariness of $\epsilon$. ∎

Lemma 3.2**.**

Let $p\geq 1$, $\gamma>1$ $\sigma=\sigma(n,p,\Lambda,\epsilon,M,\omega_{M})\geq 6$ and $\epsilon>0$ sufficiently small. Let $\{B_{r}(y_{i})\}_{i=1}^{L}$ be a sequence of balls with centers $y_{i}\in\overline{\Omega}$ and a common radius $0<r<\frac{R}{400\sigma}$ Then there exists sufficiently large number $N=N(n,p,\Lambda)>1$ and some positive number $\delta=\delta(n,p,\Lambda,\epsilon,\gamma,M,\omega_{M})>0$, such that the following statement holds. Suppose that $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (1.1) with $\|u\|_{L^{\infty}(\Omega)}\leq M$ and the nonlinearity $A(x,z,\xi)$ satisfies (1.6). If $\Omega$ is a $(\delta,R)$-Reifenberg flat domain and the following inequality holds

[TABLE]

for some $\omega\in A_{q}$, $q>1$ and $[\omega]_{q}\leq\gamma$. Then, we have

[TABLE]

where $\epsilon_{1}$ is defined in Lemma2.6

Proof.

Let $N$, $\delta$ be defined as in Lemma3.1, let

[TABLE]

and

[TABLE]

by applying Lemma2.6 and Lemma3.1, we can complete the proof of the Lemma. ∎

Corollary 3.3**.**

Let $p\geq 1$, $\gamma>1$ and let $\Omega,\{B_{r}(y_{i})\}_{i=1}^{L},\epsilon,N,\delta$ be as in Lemma3.2. Suppose that $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (1.1) with $\|u\|_{L^{\infty}(\Omega)}\leq M$ and the nonlinearity $A(x,z,\xi)$ satisfies (1.6). If

[TABLE]

for some $\omega\in A_{q}$, $q>1$ and $[\omega]_{q}\leq\gamma$. For $\forall\beta>0$, set $\epsilon_{2}=\max\{1,2^{\beta-1}\}\epsilon_{1}^{\beta}$, where $\epsilon_{1}$ is defined in Lemma2.6, then we have

[TABLE]

Proof.

We now prove this corollary by induction. The case $k=1$ follows from Lemma3.2, suppose now that the conclusion is true for some $k>1$. Let $u_{N}=\frac{u}{N}$ and $f_{N}=\frac{f}{N}$, we discover that

[TABLE]

for $i=1,\cdots,L$. Where the second inequality holds because of $N>1$ and the last one is due to assumption (3.11). Now by induction assumption it follows that

[TABLE]

Here we have used the case $k=1$ to the first term in the forth inequality. Hence we complete the proof of the corollary. ∎

4 Weighted Lorentz estimates.

Before proving the main result, we provide some elementary estimates that will be crucial for obtaining the Calderón-Zygmund type estimates.

Lemma 4.1**.**

(cf.[16][18]). Let $p>1$ and $\Omega\subset\mathbb{R}^{n}$ be a bounded open set. Assume that $A(x,z,\xi)$ satisfies (1.3). Then for any $\xi_{1},\xi_{2}\in W^{1,p}(\Omega)$ and any nonnegative function $\phi\in C(\overline{\Omega})$, it holds that

1. (1)

If $1<p<2$, then for any $\tau>0$,

[TABLE] 2. (2)

If $p\geq 2$, then

[TABLE]

Global $L^{p}$ estimate of (1.1) is stated in the following theorem.

Lemma 4.2**.**

Assume $A(x,z,\xi)$ satisfies (1.3). Let $F\in L^{p}(\Omega,\mathbb{R}^{n})$ and $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (1.1), then

[TABLE]

Where $C=C(n,p,\Lambda)$

Proof.

Let $u$ as a test function of (1.1), we have

[TABLE]

for $\forall\tau>0$, where we used Young inequality. Applying Lemma4.1, we get

[TABLE]

Choose $\tau=\frac{1}{2C^{*}}$, we have

[TABLE]

∎

With these preliminary estimates at hand, we may now proceed to the proof of the weighted regularity estimate.

Proof of Theorem 1.5.

We will consider only the case $t\neq\infty$, as for $t=\infty$, the proof is similar. Let $N=N(n,p,\Lambda)$ be defined as in Corollary3.3. For $q>1$, take $\epsilon_{1}=\epsilon\left(\frac{10}{1-4\delta}\right)^{nq}[\omega]_{q}^{2}$, $\epsilon_{2}=max\left\{1,2^{\frac{t}{pq}-1}\right\}\epsilon_{1}^{\frac{t}{pq}}$, choose $\epsilon$ sufficiently small such that

[TABLE]

Let $\delta=\delta(n,p,\Lambda,\epsilon,\gamma)$ is determined by Corollary3.3. Assume that the assumptions of Theorem1.5 hold with this choice of $\delta$. Furthermore, assume that $u$ is a weak solution of (1.1), we select a finite collection of points $\{y_{i}\}_{i=1}^{L}\subset\overline{\Omega}$ and a ball $B$ such that $\overline{\Omega}\subset\cup_{i=1}^{L}B_{r}(y_{i})\subset B$, where $r=\frac{R}{400\sigma}$. We now prove Theorem1.5 with the following additional assumption that

[TABLE]

Where $\mu=\tilde{C}\|\nabla u\|_{L^{p}(\Omega)}$ with some sufficiently large constant $\tilde{C}$ depending on $n,p,q,\Lambda,\gamma,\Omega,\epsilon$ which is to be determined later. For $t\neq\infty$, we now consider the sum

[TABLE]

Let $\Gamma=N^{p}>1$, then

[TABLE]

Owing to (4.2) and applying Corollary3.3, take $\beta=\frac{t}{pq}$ we have

[TABLE]

To control $S$, we employ Fubini’s theorem and Lemma2.3 to calculate:

[TABLE]

where $F_{\mu}=\frac{F}{\mu}$. Note that the choice of $\epsilon_{2}$, applying the Lemma2.3 again, we obtain

[TABLE]

for a constant $C$ depending on $n,p,\Lambda,t$, where $u_{\mu}=\frac{u}{\mu}$. Also, by the Lebesgue’s differentiation theorem and the definition of weighted Lorentz space, we see that

[TABLE]

Using the last inequality and Lemma2.5, we obtain

[TABLE]

Owing to the definition of $\mu$ and Lemma4.2, we get that

[TABLE]

By appealing to Lemma2.2, we get that there exists a constant $s=s(n,q,\gamma)$ such that $q-s>1$ and $\omega\in A_{q-s}$ with $[\omega]_{q-s}\leq C(n,q,\gamma)$. Hence, we can estimate $\|F\|^{p}_{L^{p}(\Omega)}$ as follows.

[TABLE]

Where we used Hölder inequality and embedding theorem as mentioned in Lemma2.4. Plugging this and (4.9) into (4.8), we end up with

[TABLE]

Summarizing the efforts, we complete the proof of the Theorem as long as we can prove (4.2). Let

[TABLE]

Owing to Lemma2.1, we have the following estimates.

[TABLE]

Where $\alpha$ is the constant as in Lemma2.1. Then by weak (1,1)-type estimate for maximal functions, there exists a constant such that

[TABLE]

It follows that

[TABLE]

Now, we choose $\widetilde{C}$ sufficiently large such that

[TABLE]

which gives estimate (4.2) as desired. ∎

5 Besov regularity for solutions of a class of special harmonic equations.

In this section, we study the Besov regularity for solutions of (1.8), in the process, Calderón-Zygmund estimate will play an important role. For the sake of convenience and simplicity, we take advantage of Calderón-Zygmund estimate in a special case of $F=0$, $p=2$, $t=q$, $\omega=1$ and $\omega_{M}(t)=t^{\alpha}$. In this case, (1.2) and (1.3) can be rewritten as

[TABLE]

[TABLE]

and

[TABLE]

Given a domain $\Omega\subset\mathbb{R}^{n}$, we say that $f$ belongs to the local Besov space $B_{p,q,loc}^{\alpha}$ if $\varphi f$ belongs to the global Besov space $B_{p,q}^{\alpha}(\mathbb{R}^{n})$ for any $\varphi\in C_{0}^{\infty}(\Omega)$. Besides, we have the following technical lemma (cf.[6]).

Lemma 5.1**.**

A function $f\in L_{loc}^{p}(\Omega)$ belongs to the local Besov space $B_{p,q,loc}^{\alpha}$ if and only if

[TABLE]

for any ball $B\subset 2B\subset\Omega$ with radius $r_{B}$. Where $\Delta_{h}f(x)=f(x+h)-f(x)$. Here the measure $\frac{\operatorname{d}\!h}{|h|^{n}}$ is restricted to the ball $B(0,r_{B})$ on the $h$-space.

Next, we introduce some elementary estimates.

Lemma 5.2**.**

Suppose $1\leq p<\infty$, $u\in W^{1,p}(B_{R})$. Then, for each $0<\rho<R$, we have

[TABLE]

for all $0<|h|<\frac{R-\rho}{2}$.

Lemma 5.3**.**

Let $A(x,z,\xi)$ satisfies (1.7), (5.1)-(5.3). Then $A(x,z,\xi)$ has small BMO semi-norm in $x$, i.e. (1.6) holds.

Proof.

[TABLE]

Where we used Hölder inequality. Thus, owing to the absolute continuity of the integral, we complete the proof. ∎

Now we proceed by proving Theorem 1.9

Proof of Theorem 1.9.

Fix a ball $B_{R}$ such that $B_{2R}\subset\subset\Omega$. Let $\eta\in C_{0}^{\infty}(B_{R})$ with $\eta=1$ on $B_{\frac{R}{2}}$ and $|\nabla\eta|\leq\frac{C}{R}$. For small enough $|h|$, given a test function $\varphi=\Delta_{-h}\left(\eta^{2}\triangle_{h}u\right)$, we test the equation(1.8) with $\varphi$, we have

[TABLE]

Combine this and the “integration-by-part” formula for difference quotients, we get

[TABLE]

We can write (5.4) as follows:

[TABLE]

Taking advantage of (5.1) in the left-hand side, we have

[TABLE]

Now, we estimate $I_{1}$-$I_{5}$ respectively. We proceed by estimating $I_{1}$ from (5.2) that

[TABLE]

We use (5.3) and Young inequality as follows:

[TABLE]

and

[TABLE]

By virtue of assumption (1.7) and Young inequality, we have

[TABLE]

and

[TABLE]

Collecting the above estimates, we get

[TABLE]

From Lemma5.2 and the fact that $|\nabla\eta|\leq\frac{C}{R}$, the first term on the right-hand side can be estimated as:

[TABLE]

Owing to Hölder inequality and Lemma5.2, we obtain

[TABLE]

and

[TABLE]

The homogeneity of the equation together with Calderón-Zygmund estimate yield that $\nabla u\in L_{loc}^{s}(\Omega)$ for $\forall s>1$, see Theorem1.6 with $F=0$, $p=2$, $t=q$, $\omega=1$. In particular, $\nabla u\in L^{\frac{2}{1-\alpha}}(B_{R})$ and $\nabla u\in L^{\frac{2n}{n-2\alpha}}(B_{R})$. Thus, from Hölder inequality, we have

[TABLE]

Combining all this estimates and divide both side of (5.5) by $|h|^{2\alpha}$. Moreover, we use the fact that $\eta=1$ on $B_{\frac{R}{2}}$, then

[TABLE]

Now, we take supremum over all $h\in B_{\delta}$ for some $\delta<R$. Since $g\in L_{loc}^{\frac{n}{\alpha}}(\Omega)$, the proof of Theorem1.9 is complete. ∎