A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight

Abstract

This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the $\mathcal{A}$-harmonic equation $\mathsf{\text{div}}\mathcal{A}\left(x,\nabla u\right)=0$ with $\left|\mathcal{A}\left(x,\xi \right)\right|\approx w\left(x\right)|\xi {|}^{p-1}$, where 1 < p < ∞ and w(x) be a Muckenhoupt A₁ weight.

Mathematics Subject Classification (2000) 35J50, 35J60

1 Introduction

Let w be a locally integrable non-negative function in Rⁿ and assume that 0 < w < ∞ almost everywhere. We say that w belongs to the Muckenhoupt class A _p , 1 < p < ∞, or that w is an A _p weight, if there is a constant A _p (w) such that

$$\underset{B}{sup}\left(\frac{1}{\left|B\right|}\underset{B}{\int }w\mathsf{\text{d}}x\right){\left(\frac{1}{\left|B\right|}\underset{B}{\int }{w}^{1∕\left(1-p\right)}\mathsf{\text{d}}x\right)}^{p-1}=A_p\left(w\right)<\infty$$

(1.1)

for all balls B in Rⁿ. We say that w belongs to A₁, or that w is an A₁ weight, if there is a constant A₁(w) such that

$$\frac{1}{\left|B\right|}\underset{B}{\int }w\mathsf{\text{d}}x\le A_1\left(w\right)\mathsf{\text{essin}}{\mathsf{\text{f}}}_{B}w$$

for all balls B in Rⁿ.

As customary, μ stands for the measure whose Radon-Nikodym derivative w is

$$\mu \left(E\right)=\underset{E}{\int }w\mathsf{\text{d}}x.$$

It is well known that A₁ ⊂ A _p whenever p > 1, see [1]. We say that a weight w is doubling if there is a constant C > 0 such that

$$\mu \left(2B\right)\le C\mu \left(B\right)$$

whenever B ⊂ 2B are concentric balls in Rⁿ, where 2B is the ball with the same center as B and with radius twice that of B. Given a measurable subset E of Rⁿ, we will denote by L^p(E, w), 1 < p < ∞, the Banach space of all measurable functions f defined on E for which

$$\left|\right|f|{|}_{L^p(E,w)}={\left({\displaystyle \underset{E}{\int }}|f(x{)|}^{p}w(x)\text{d}x\right)}^{\frac{1}{p}}<\infty \mathrm{.}$$

The weighted Sobolev class W^1,p(E, w) consists of all functions f, and its first generalized derivatives belong to L^p (E, w). The symbols $L_{\mathsf{\text{loc}}}^p\left(E,w\right)$ and $W_{\mathsf{\text{loc}}}^{1,p}\left(E,w\right)$ are self-explanatory.

If x₀ ∈ Ω and t > 0, then B _t denotes the ball of radius t centered at x₀. For the function u(x) and k > 0, let A _k = {x ∈ Ω : |u(x)| > k}, A_k,t= A _k ∩ B _t . Let T _k (u) be the usual truncation of u at level k > 0, that is

$$T_k\left(u\right)=max\left\{-k,min\left\{k,u\right\}\right\}.$$

Let Ω be a bounded regular domain in Rⁿ, n ≥ 2. By a regular domain, we understand any domain of finite measure for which the estimates for the Hodge decomposition in (2.1) and (2.2) are satisfied. A Lipschitz domain, for example, is regular. We consider the second-order degenerate elliptic equation (also called $\mathcal{A}$-harmonic equation or Leray-Lions equation)

$$\mathsf{\text{div}}\mathcal{A}\left(x,\nabla u\right)=0$$

(1.2)

where $\mathcal{A}\left(x,\xi \right):\Omega \times R^n\to R^n$ is a carathéodory function satisfying the following assumptions

• 1. $⟨\mathcal{A}\left(x,\xi \right),\xi ⟩\ge \alpha w\left(x\right)|\xi {|}^p$,

• 2. $\left|\mathcal{A}\left(x,\xi \right)\right|\le \beta w\left(x\right)|\xi {|}^{p-1}$,

where 0 < α ≤ β < ∞, w ∈ A₁ and w ≥ k₀ > 0. Suppose ψ is any function in Ω with values in the extended reals [-∞, +∞] and that θ ∈ W^1,r(Ω, w), max{1, p -1} < r ≤ p. Let

$${\mathcal{K}}_{\psi ,\theta }^r={\mathcal{K}}_{\psi ,\theta }^r\left(\Omega ,w\right)=\left\{v\in W^{1,r}\left(\Omega ,w\right):v\ge \psi ,\mathsf{\text{a.e.}}x\in \Omega \mathsf{\text{and}}v-\theta \in W_0^{1,r}\left(\Omega ,w\right)\right\}.$$

The function ψ is an obstacle, and θ determines the boundary values.

We introduce the Hodge decomposition for $|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)\in L^{\frac{r}{r-p+1}}\left(\Omega ,w\right)$, from Lemma 1 in Section 2,

$$|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)=\nabla \phi +H$$

(1.3)

and the following estimate holds

$$\parallel H{\parallel }_{L^{\frac{r}{r-p+1}}\left(\Omega ,w\right)}\le c{A}_p{\left(w\right)}^{\gamma }|r-p|\parallel \nabla \left(v-u\right){\parallel }_{L^r\left(\Omega ,w\right)}^{r-p+1}.$$

(1.4)

Definition 1 A very weak solution to the ${\mathcal{K}}_{\psi ,\theta }^r$ -obstacle problem is a function $u\in {\mathcal{K}}_{\psi ,\theta }^r\left(\Omega ,w\right)$ such that

$$\underset{\Omega }{\int }⟨\mathcal{A}\left(x,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩\mathsf{\text{d}}x\ge \underset{\Omega }{\int }⟨\mathcal{A}\left(x,\nabla u\right),H⟩\mathsf{\text{d}}x$$

(1.5)

whenever $v\in {\mathcal{K}}_{\psi ,\theta }^r\left(\Omega ,w\right)$ and H comes from the Hodge decomposition (1.3).

The local and global higher integrability of the derivatives in obstacle problems with w(x) ≡ 1 was first considered by Li and Martio [2] in 1994, using the so-called reverse Hölder inequality. Gao and Tian [3] gave a local regularity result for weak solutions to obstacle problem in 2004. Recently, regularity theory for very weak solutions of the $\mathcal{A}$-harmonic equations with w(x) ≡ 1 have been considered [4], and the regularity theory for very solutions of obstacle problems with w(x) ≡ 1 have been explored in [5]. This paper gives a Caccioppoli-type estimate for solutions to obstacle problems with weight, which is closely related to the local regularity theory for very weak solutions of the $\mathcal{A}$-harmonic equation (1.2).

Theorem There exists r₁ ∈ (p - 1, p) such that for arbitrary $\psi \in W_{\mathsf{\text{loc}}}^{1,p}\left(\Omega ,w\right)$ and r₁ < r < p, a solution u to the ${\mathcal{K}}_{\psi ,\theta }^r$-obstacle problem with weight w(x) ∈ A₁ satisfies the following Caccioppoli-type estimate

$$\underset{A_{k,\rho }}{\int }|\nabla u{|}^{r}d\mu \le C\left[\underset{A_{k,R}}{\int }|\nabla \psi {|}^{r}d\mu +\frac{1}{{\left(R-\rho \right)}^r}\underset{A_{k,R}}{\int }|u{|}^r\mathsf{\text{d}}\mu \right]$$

where 0 < ρ < R < +∞ and C is a constant depends only on n, p and β/α.

2 Preliminary Lemmas

The following lemma comes from [6] which is a Hodge decomposition in weighted spaces.

Lemma 1 Let Ω be a regular domain of Rⁿ and w(x) be an A₁ weight. If $u\in W_0^{1,p-\epsilon }\left(\Omega ,w\right)$, 1 < p < ∞, -1 < ε < p - 1, then there exist $\phi \in W_0^{1,\frac{p-\epsilon }{1-\epsilon }}\left(\Omega ,w\right)$ and a divergence-free vector field $H\in L^{\frac{p-\epsilon }{1-\epsilon }}\left(\Omega ,w\right)$ such that

$$|\nabla u{|}^{-\epsilon }\nabla u=\nabla \phi +H$$

and

$$\parallel \nabla \phi {\parallel }_{L^{\frac{p-\epsilon }{1-\epsilon }}\left(\Omega ,w\right)}\le c{A}_p{\left(w\right)}^{\gamma }\parallel \nabla u{\parallel }_{L^{p-\epsilon }\left(\Omega ,w\right)}^{1-\epsilon }$$

(2.1)

$$\parallel H{\parallel }_{L^{\frac{p-\epsilon }{1-\epsilon }}\left(\Omega ,w\right)}\le c{A}_p{\left(w\right)}^{\gamma }\left|\epsilon \right|\parallel \nabla u{\parallel }_{L^{p-\epsilon }\left(\Omega ,w\right)}^{1-\epsilon }$$

(2.2)

where γ depends only on p.

We also need the following lemma in the proof of the main theorem.

Lemma 2 [7] Let f(t) be a non-negative bounded function defined for 0 ≤ T₀ ≤ t ≤ T₁. Suppose that for T₀ ≤ t < s ≤ T₁, we have

$$f\left(t\right)\le A{\left(s-t\right)}^{-\alpha }+B+\theta f\left(s\right),$$

where A, B, α, θ are non-negative constants and θ < 1. Then, there exist a constant c, depending only on α and θ, such that for every ρ, R, T₀ ≤ ρ < R ≤ T₁ we have

$$f\left(\rho \right)\le c\left[A{\left(R-\rho \right)}^{-\alpha }+B\right].$$

3 Proof of the main theorem

Let u be a very weak solution to the ${\mathcal{K}}_{\psi ,\theta }^r$-obstacle problem. Let $B_{R_1}\subset \subset \Omega$ and 0 < R₀ ≤ τ < t ≤ R₁ be arbitrarily fixed. Fix a cut-off function $\varphi \in C_0^{\infty }\left(B_t\right)$ such that

$$\text{supp}\varphi \subset B_t,0\le \varphi \le 1,\varphi =1\text{in}{B}_{\tau }\text{and}|\nabla \varphi |\le 2(t-\tau {)}^{-1}.$$

Consider the function

$$v=u-T_k\left(u\right)-{\varphi }^r\left(u-{\psi }_k^{+}\right),$$

where T _k (u) is the usual truncation of u at the level k defined in Section 1 and ${\psi }_k^{+}=max\left\{\psi ,T_k\left(u\right)\right\}$. Now $v\in {\mathcal{K}}_{\psi -T_k\left(u\right),\theta -T_k\left(u\right)}^r\left(\Omega ,w\right)$. Indeed,

$$v-\left(\theta -T_k\left(u\right)\right)=u-\theta -{\varphi }^r\left(u-{\psi }_k^{+}\right)\in W_0^{1,r}\left(\Omega ,w\right)$$

since $\varphi \in C_0^{\infty }\left(\Omega \right)$ and

$$v-\left(\psi -T_k\left(u\right)\right)=\left(u-\psi \right)-{\varphi }^r\left(u-{\psi }_k^{+}\right)\ge \left(1-{\varphi }^r\right)\left(u-\psi \right)\ge 0$$

a.e. in Ω. Let

$$E\left(v,u\right)=|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u+|\nabla \left(v-u+T_k\left(u\right)\right){|}^{r-p}\nabla \left(v-u+T_k\left(u\right)\right).$$

(3.1)

From an elementary formula [[8], (4.1)]

$$\left|\right|X{|}^{-\epsilon }X-\left|Y{|}^{-\epsilon }Y\right|\le 2^{\epsilon }\frac{1+\epsilon }{1-\epsilon }|X-Y{|}^{1-\epsilon },X,Y\in {\mathsf{\text{R}}}^n,0\le \epsilon <1$$

and $\nabla v=\nabla \left(u-T_k\left(u\right)\right)-{\varphi }^r\nabla \left(u-{\psi }_k^{+}\right)-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k^{+}\right)$, we can derive that

$$\left|E\left(v,u\right)\right|\le 2^{p-r}\frac{p-r+1}{r-p+1}|{\varphi }^r\nabla u-{\varphi }^r\nabla \left(u-{\psi }_k^{+}\right)-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k^{+}\right){|}^{r-p+1}.$$

(3.2)

From (3.1), we get that

$$\begin{array}{lll}\hfill \underset{A_{k,t}}{\int }⟨\mathcal{A}\left(x,\nabla u\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩\mathsf{\text{d}}x=& \underset{A_{k,t}}{\int }⟨\mathcal{A}\left(x,\nabla u\right),E\left(v,u\right)⟩\mathsf{\text{d}}x& \hfill \text{(1)}\\ -\underset{A_{k,t}}{\int }⟨\mathcal{A}\left(x,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩\mathsf{\text{d}}x.& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

(3.3)

Now we estimate the left-hand side of (3.3),

$$\underset{A_{k,t}}{\int }⟨A\left(x,\nabla u\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩\mathsf{\text{d}}x\ge \underset{A_{k,\tau }}{\int }⟨A\left(x,\nabla u\right),|\nabla u{|}^{r-p}\nabla u⟩\mathsf{\text{d}}x\ge \alpha \underset{A_{k,\tau }}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu .$$

(3.4)

Using (1.3), we get

$$|\nabla \left(v-u+T_k\left(u\right)\right){|}^{r-p}\nabla \left(v-u+T_k\left(u\right)\right)=\nabla \phi +H$$

(3.5)

and (1.4) yields

$$\parallel H{\parallel }_{L^{\frac{r}{r-p+1}}\left(\Omega ,w\right)}\le c{A}_p{\left(w\right)}^{\gamma }|r-p|\parallel \nabla \left(v-u+T_k\left(u\right)\right){\parallel }_{L^r\left(\Omega ,w\right)}^{r-p+1}.$$

(3.6)

Since u - T _k (u) is a very weak solution to the ${\mathcal{K}}_{\psi -T_k\left(u\right),\theta -T_k\left(u\right)}^r$-obstacle problem, we derive, by

Definition 1, that

$$\underset{\Omega }{\int }⟨\mathcal{A}\left(x,\nabla \left(u-T_k\left(u\right)\right)\right),|\nabla \left(v-u+T_k\left(u\right)\right){|}^{r-p}\nabla \left(v-u+T_k\left(u\right)\right)⟩\mathsf{\text{d}}x\ge \underset{\Omega }{\int }⟨\mathcal{A}\left(x,\nabla \left(u-T_k\left(u\right)\right)\right),H⟩\mathsf{\text{d}}x$$

that is

$$\underset{A_{k,t}}{\int }⟨\mathcal{A}\left(x,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩\mathsf{\text{d}}x\ge \underset{A_{k,t}}{\int }⟨\mathcal{A}\left(x,\nabla u\right),H⟩\mathsf{\text{d}}x.$$

(3.7)

Combining the inequalities (3.3), (3.4) and (3.7), we obtain

$$\begin{array}{c}\alpha {\displaystyle \underset{A_{k,\tau }}{\int }}|\nabla u{|}^r\text{d}\mu \le {\displaystyle \underset{A_{k,t}}{\int }}〈A(x,\nabla u),E(v,u)〉\text{d}x-{\displaystyle \underset{A_{k,t}}{\int }}〈A(x,\nabla u),H〉\text{d}x\\ \le \beta \frac{2^{p-r}(p-r+1)}{r-p+1}{\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^{p-1}|{\varphi }^r\nabla {\psi }_k^{+}-r{\varphi }^{r-1}\nabla \varphi (u-{\psi }_k^{+}){|}^{r-p+1}\text{d}\mu \\ +\beta {\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^{p-1}|H|\text{d}\mu \\ \le \beta \frac{2^{p-r}(p-r+1)}{r-p+1}{\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^{p-1}|{\varphi }^r\nabla \psi {|}^{r-p+1}\text{d}\mu \\ +\beta \frac{2^{p-r}(p-r+1)}{r-p+1}{\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^{p-1}|r{\varphi }^{r-1}\nabla \varphi (u-{\psi }_k^{+}){|}^{r-p+1}\text{d}\mu \\ +\beta {\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^{p-1}|H|\text{d}\mu \\ \le \beta \frac{2^{p-r}(p-r+1)}{r-p+1}{\left({\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^r\text{d}\mu \right)}^{\frac{p-1}{r}}{\left({\displaystyle \underset{A_{k,t}}{\int }}|\nabla \psi {|}^r\text{d}\mu \right)}^{\frac{r-p+1}{r}}\\ +\beta \frac{2^{p-r}(p-r+1)}{r-p+1}{\left({\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^r\text{d}\mu \right)}^{\frac{p-1}{r}}{\left({\displaystyle \underset{A_{k,t}}{\int }}|r{\varphi }^{p-1}\nabla \varphi (u-{\psi }_k^{+}{)|}^r\text{d}\mu \right)}^{\frac{r-p+1}{r}}\\ +\beta {\left({\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^r\text{d}\mu \right)}^{\frac{p-1}{r}}{\left({\displaystyle \underset{A_{k,t}}{\int }}|H{|}^{\frac{r}{r-p+1}}\text{d}\mu \right)}^{\frac{r-p+1}{r}}\mathrm{.}\end{array}$$

Let $c_1=\frac{2^{p-r}\left(p-r+1\right)}{r-p+1}$, by (3.6) and Young's inequality

$$ab\le \epsilon a^{p^{\prime }}+c_2\left(\epsilon ,p\right)b^p,\frac{1}{p^{\prime }}+\frac{1}{p}=1,a,b\ge 0,\epsilon \ge 0,p\ge 1,$$

we can derive that

$$\begin{array}{lll}\hfill \alpha \underset{A_{k,\tau }}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu \le & \beta c_1\epsilon \underset{A_{k,t}}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu +\beta c_1{c}_2\left(\epsilon ,p\right)\underset{A_{k,t}}{\int }|\nabla \psi {|}^r\mathsf{\text{d}}\mu & \hfill \text{(1) }\\ +\beta c_1\epsilon \underset{A_{k,t}}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu +\beta c_1{c}_2\left(\epsilon ,p\right)\underset{A_{k,t}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k^{+}\right){|}^r\mathsf{\text{d}}\mu & \hfill \text{(2) }\\ +\beta c{A}_p{\left(w\right)}^{\gamma }\left(p-r\right)\epsilon \underset{A_{k,t}}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu & \hfill \text{(3) }\\ +\beta c{A}_p{\left(w\right)}^{\gamma }\left(p-r\right)c_2\left(\epsilon ,p\right)\underset{\Omega }{\int }|\nabla \left(v-u+T_k\left(u\right)\right){|}^r\mathsf{\text{d}}\mu ,& \hfill \text{(4) }\\ \hfill \text{(5) }\end{array}$$

where c is the constant given by Lemma 1. Since v - u + T _k (u) = 0 on Ω\A_k,t, by the equality

$$\nabla v=\nabla \left(u-T_k\left(u\right)\right)-{\varphi }^r\nabla \left(u-{\psi }_k^{+}\right)-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k^{+}\right),$$

we obtain that

$$\begin{array}{c}\underset{\Omega }{\int }|\nabla \left(v-u+T_k\left(u\right)\right){|}^r\mathsf{\text{d}}\mu =\underset{A_{k,t}}{\int }|\nabla \left(v-u\right){|}^r\mathsf{\text{d}}\mu \\ =\underset{A_{k,t}}{\int }|{\varphi }^r\nabla \left(u-{\psi }_k^{+}\right)+r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k^{+}\right){|}^r\mathsf{\text{d}}\mu \\ \le 2^{r-1}\underset{A_{k,t}}{\int }|\nabla \left(u-{\psi }_k^{+}\right){|}^r\mathsf{\text{d}}\mu +2^{r-1}r\underset{Ak,t}{\int }|\nabla \varphi \left(u-{\psi }_k^{+}\right){|}^r\mathsf{\text{d}}\mu \\ \le 2^{2r-2}\underset{A_{k,t}}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu +2^{2r-2}\underset{A_{k,t}}{\int }|\nabla \psi {|}^r\mathsf{\text{d}}\mu +r{2}^{2r-2}\underset{A_{k,t}}{\int }\frac{\left|u^r\right|}{{\left(t-\tau \right)}^r}\mathsf{\text{d}}\mu .\end{array}$$

Finally, we obtain

$$\begin{array}{c}{\displaystyle \underset{A_{k,\tau }}{\int }}|\nabla u{|}^r\text{d}\mu \le \frac{\beta (2c_1+c{A}_p{(w)}^{\gamma }(p-r))\epsilon +\beta c{A}_p{(w)}^{\gamma }{c}_2(\epsilon ,p{)2}^{2r-2}(p-r)}{\alpha }{\displaystyle \underset{A_{k,t}}{\int }}|\nabla u{|}^r\text{d}\mu \\ +\frac{\beta c_1{c}_2(\epsilon ,p)+2^{2r-2}\beta c{A}_p{(w)}^{\gamma }{c}_2(\epsilon ,p)(p-r)}{\alpha }{\displaystyle \underset{A_{k,t}}{\int }}|\nabla \psi {|}^r\text{d}\mu \\ +r\frac{\beta c_1{c}_2(\epsilon ,p)+2^{2r-1}\beta c{A}_p{(w)}^{\gamma }{c}_2(\epsilon ,p)(p-r)}{\alpha }{\displaystyle \underset{A_{k,t}}{\int }}\frac{|u{|}^r}{{(t-\tau )}^r}\text{d}\mu \mathrm{.}\end{array}$$

(3.8)

Now we want to eliminate the first term in the right-hand side containing ∇u. Choosing ε and r₁ such that

$$\eta =\frac{\beta (2c_1+c{A}_p{(w)}^{\gamma }(p-r))\epsilon +\beta c{A}_p{(w)}^{\gamma }{c}_2(\epsilon ,p{)2}^{2r-2}(p-r)}{\alpha }<1$$

and let ρ, R be arbitrarily fixed with R₀ ≤ ρ < R ≤ R₁. Thus, from (3.8), we deduce that for every t and τ such that ρ ≤ τ < t ≤ R, we have

$$\underset{A_{k,\tau }}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu \le \eta \underset{A_{k,t}}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu +\frac{c_3}{\alpha }\underset{A_{k,t}}{\int }|\nabla \psi |\mathsf{\text{d}}\mu +\frac{c_4}{\alpha {\left(t-\tau \right)}^r}\underset{A_{k,t}}{\int }|u{|}^r\mathsf{\text{d}}\mu ,$$

(3.9)

where

$$c_3=\beta c_1{c}_2\left(\epsilon ,p\right)+2^{2r-2}\beta c{A}_p{\left(w\right)}^{\gamma }{c}_2\left(\epsilon ,p\right)\left(p-r\right)$$

and

$$c_4=r\beta c_1{c}_2\left(\epsilon ,p\right)+r{2}^{2r-1}\beta c{A}_p{\left(w\right)}^{\gamma }{c}_2\left(\epsilon ,p\right)\left(p-r\right).$$

Applying Lemma 2 in (3.9), we conclude that

$$\underset{A_{k,\rho }}{\int }|\nabla u{|}^r\mathsf{\text{d}}\mu \le \frac{c{c}_3}{\alpha }\underset{A_{k,R}}{\int }|\nabla \psi {|}^{r}d\mu +\frac{c{c}_4}{\alpha {\left(R-\rho \right)}^r}\underset{A_{k,R}}{\int }|u{|}^r\mathsf{\text{d}}\mu ,$$

where c is the constant given by Lemma 2. This ends the proof of the main theorem.