Global BMO Estimation for the Gradient of Weak Solutions to a Class of Elliptic Obstacle Problems

Tong Yuxia, Guo Yanmin, Gu Jiantao,*

College of Science, North China University of Science and Technology, Hebei Tangshan 063210

 基金资助: 河北省教育厅重点项目(ZD2022070)

 Fund supported: key project of Hebei Provincial Department of Education(ZD2022070)

Abstract

In this paper, the global BMO estimates for the gradient of weak solutions to a class of elliptic equation obstacle problems is considered by using the Hardy-Littlewood maximal functions, the Jensen inequality for Young function, the perturbation method and other techniques.

Keywords： Obstacle problem; Weak solution; BMO estimation

Tong Yuxia, Guo Yanmin, Gu Jiantao. Global BMO Estimation for the Gradient of Weak Solutions to a Class of Elliptic Obstacle Problems[J]. Acta Mathematica Scientia, 2023, 43(1): 159-168

1 引言

${\rm div} (a(|\nabla u|)\nabla u)=f(x),\qquad x\in \Bbb R^n$

$0\leq i_{a}:=\inf_{t>0}\frac{ta'(t)}{a(t)}\leq \sup_{t>0}\frac{ta'(t)}{a(t)}=:s_{a}<\infty$

${\rm div} (a(|\nabla u|)\nabla u)={\rm div}{\boldsymbol f},$

$$$\label{(1.1)} {\rm div} (a(|\nabla u|)\nabla u)={\rm div}(a(|{\boldsymbol f}|){\boldsymbol f})$$$

$$$\label{(1.2)} 0\leq i_{a}:=\inf_{t>0}\frac{ta'(t)}{a(t)}\leq \sup_{t>0}\frac{ta'(t)}{a(t)}=:s_{a}<\infty.$$$

${\rm div} (|\nabla u|^{p-2}\nabla u)= {\rm div} (|{\boldsymbol f}|^{p-2}{\boldsymbol f}).$

$$$\label{(1.3)} b(t)=ta(t),\quad B(t)=\int _{0}^{t}\tau a(\tau) {\rm d}\tau=\int _{0}^{t}b(\tau) {\rm d}\tau, \quad \mbox{当}\ t\geq0.$$$

$\psi$${\Bbb R}^n 上取值于 \Bbb R\bigcup \{\pm\infty\} 的任意函数, 定义障碍问题的容许函数类 \begin{eqnarray*} K_{\psi}^{B}({\Bbb R}^n ):=\left \{u \in W^{1,B}({\Bbb R}^n ): \mbox{在 {\Bbb R}^n 上几乎处处有}\ u\geq\psi\right \}. \end{eqnarray*} 关于 W^{1,B}({\Bbb R}^n) 的含义见下节. 首先给出如下定义. 定义1.1 函数 u \in K_{\psi}^{B} ({\Bbb R}^n) 称为方程 (1.1)$$K_{\psi}^{B}$ - 障碍问题的弱解, 若

$$$\label{(1.4)} \int _{{\Bbb R}^n }\left \langle a(|\nabla u|)\nabla u,\nabla (v-u) \right \rangle {\rm d}x \geq \int _{{\Bbb R}^n}\left \langle a(| {\boldsymbol f} |) {\boldsymbol f},\nabla (v-u) \right \rangle {\rm d}x$$$

DiBenedetto 和 Manfredi[6]$p>2$ 的情况下获得了拟线性椭圆方程组 ${\rm div} (|Du|^{p-2}Du)\\ ={\rm div}(|{\boldsymbol F}|^{p-2}{\boldsymbol F})$ 弱解梯度在全空间 ${\Bbb R}^n$上的全局 BMO 估计; Diening, Kaplický 和 Schwarz-acher[7]$p>1$ 的情况下获得了非齐次椭圆方程组 $-{\rm div} (A(\nabla u))={\rm div}\textbf{f} $${\Bbb R}^n 中的球 B 上的局部 BMO 估计; Yao, Zhang 和 Zhou[3] 利用Hardy-Littlewood极大函数有界性获得了一类拟线性椭圆方程 {\rm div} (a(|\nabla u|)\nabla u)={\rm div}{\boldsymbol f} 弱解在全空间 {\Bbb R}^n 上的全局 BMO 估计; Liang 和 Zheng[8] 获得了具有部分BMO的椭圆障碍问题在 Orlicz 空间中的全局梯度估计. 关于椭圆方程的更多研究具体参见文献[9-15]. 本文主要受到 Yao, Zhang 和 Zhou 在文献[3]以及 Liang 和 Zheng 在文献[8]中关于 BMO 估计以及障碍问题的处理思想的启发, 对梯度 Du 在有限球上的局部 BMO 半范作逐点估计, 由于估计界一致性, 对半径取极限, 从而建立全空间上具有散度形式的椭圆方程障碍问题弱解梯度的 BMO 估计. 下面是本文的主要结论. 定理1.1 假设 a(t) 满足(1.2)式, B(t) 满足(1.3)式, 且有 a(|\nabla\psi|)\nabla\psi\in {\rm BMO}(\Bbb R^{n}),\quad a(|{\boldsymbol f}|){\boldsymbol f}\in {\rm BMO}(\Bbb R^{n}). u 是方程(1.1)的 K_{\psi}^{B} - 障碍问题的弱解, 则 \nabla u \in {\rm BMO}(\Bbb R^{n}), 且有估计式 \begin{matrix}\label{jieguo1.9} B\left(||\nabla u||_{{\rm BMO}(\Bbb R^{n})}\right) &\leq& C\widetilde{B}\left(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(\Bbb R^{n})}\right)+C\widetilde{B}\left(||a(|\nabla\psi|)\nabla\psi||_{ {\rm BMO}(\Bbb R^{n})}\right) \end{matrix} 成立, 其中正常数 C$$u,{\boldsymbol f},\psi$ 无关.

$\begin{matrix}\label{jieguo1.10} \|\nabla u\|_{{\rm BMO}(\Bbb R^{n})} \leq \displaystyle C\|\left | {\boldsymbol f}\right|^{p-2}{{\boldsymbol f}}\|_{{\rm BMO}(\Bbb R^{n})}^{1/(p-1)} +C\| \left | \nabla\psi\right|^{p-2}\nabla\psi\|_{{\rm BMO}(\Bbb R^{n})}^{1/(p-1)} \end{matrix}$

$$$\label{Definition 2.1} \lim_{t\rightarrow +\infty}\frac{B(t)}{t}=\lim_{t\rightarrow 0^{+} }\frac{t}{B (t)}=+\infty,$$$

$\rm {Young}$ 函数 $B$ 也称为 $N$ - 函数. 此外, 定义 $N$ - 函数 $B$$\rm {Young} 共轭 \widetilde{B} \widetilde{B}(t)=\sup_{s\geq 0}\{st-B(s)\}, \quad \mbox{当} \ t\geq0. 如果 B$$N$ - 函数, 则 $\widetilde{B}$ 也是 $N$ - 函数.

$$$\label{Definition 2.2} B(2t)\leq KB(t),$$$

$$$\label{Definition 2.3} B(t)\leq \frac{B(\theta t)}{2\theta},$$$

$\begin{eqnarray*} \int _{\Omega }B (\left | g \right |){\rm d}x< \infty \end{eqnarray*}$

$$$\label{(2.6)} \widetilde B(b(t))\leq C_{0}B(t),\quad \mbox{当}\ t\geq 0, \ C_0>0.$$$

$(2)$$B(t)\in\triangle _{2}\bigcap \bigtriangledown _{2}. 引理2.3[4] 假设 a(t) 满足(1.2)式, B(t) 满足(1.3)式, 则有 (1) 对任意 \xi\in \Bbb R^n, 有 $$\label{(2.7)} a(|\xi|)\xi\cdot\xi\geq C(i_a,s_a)B(|\xi|).$$ (2) 对任意 \xi,\eta \in\Bbb R^n, 有 $$\label{(2.8)} \left \langle a(|\xi| )\xi-a(|\eta |)\eta,\xi -\eta \right \rangle\geq C(i_a,s_a)B(|\xi-\eta|),\quad \mbox{当}\ i_a\geq 0.$$ 引理2.4[3,17]B\in\triangle _{2}, v\in W^{1,B}(B_{R}), 则有 -\!\!\!\!\!\!\int _{B_{R}}B\left( \frac{|v-v_{B_{R}}|}{R}\right){\rm d}x\leq CB\left(||v||_{{\rm BMO}(B_R)} \right). 下面给出 BMO 空间的定义. 定义2.4[3]\rm BMO 空间 BMO(\Bbb R^n) 是满足 ||h||_{BMO(\Bbb R^n)}< \infty 的所有函数 h 组成的集合, 其中, || h||_{BMO(\Bbb R^n)}:= \sup_{{x\in\Bbb R^n\atop B_{r}(x)\subset \Bbb R^n}} -\!\!\!\!\!\!\int _{B_{r}(x)}|h-h_{B_{r}(x)}|{\rm d}x. 特别地, 对于限制在球 B_R(x_0)上的 \rm BMO 半范定义为 || h||_{{\rm BMO}(B_R(x_0))}:= \sup_{0<r<R} -\!\!\!\!\!\!\int _{B_{r}(x_0)}|h-h_{B_{r}(x)}|{\rm d}x. 下面的引理在证明中起重要作用. 引理2.5[18]a(t) 满足(1.2)式, v\in W^{1,B}(B_{R}) 是方程 {\rm div} (a(|\nabla v|)\nabla v)=0 的局部弱解, 则存在两个正常数 C,\sigma, 有 $$\label{(2.9)} \sup_{{B_{R/2}}}B(|\nabla v|)\leq C -\!\!\!\!\!\!\int _{B_{R}}B(|\nabla v|){\rm d}x,$$ $$\label{(2.10)} -\!\!\!\!\!\!\int _{B_{\rho}}B(|\nabla v-(\nabla v)_{B_{\rho}}|){\rm d}x\leq C(\frac{\rho}{R})^{\sigma} -\!\!\!\!\!\!\int _{B_{R}}B(|\nabla v-(\nabla v)_{B_{R}}|){\rm d}x, \quad \mbox{对任意}\ \rho<R,$$ 其中 C,\sigma 依赖于 n,i_a,s_a. 为完成证明, 需要给出下面几个引理. w,v\in W^{1,B}(B_R) 分别为下列边值问题的弱解 \begin{matrix}\label{(2.11)} \left\{\begin{array}{ll} {\rm div} (a(|\nabla w|)\nabla w)={\rm div} (a(|\nabla \psi|)\nabla \psi),\quad & x\in B_{R},\\ w=u, &x\in\partial B_{R}. \end{array}\right. \end{matrix} \begin{matrix}\label{(2.12)} \left\{\begin{array}{ll} {\rm div} (a(|\nabla v|)\nabla v)=0,\quad& x\in B_{R},\\ v=w, & x\in\partial B_{R}. \end{array}\right. \end{matrix} 引理2.6 假设 w\in W^{1,B}(B_R) 满足问题 \begin{matrix}\label{(2.13)} \left\{\begin{array}{ll} -{\mathrm{div}} (a(|\nabla \psi|)\nabla \psi)\leq-\mathrm{div} (a(|\nabla w|)\nabla w),\quad &x\in B_{R},\\ \psi \leq w, & x\in\partial B_{R}. \end{array}\right. \end{matrix} 在弱意义下, 对所有非负函数 \varphi\in W_{0}^{1,B}(B_R) \begin{matrix}\label{(2.14)} \int_{B_R} \left \langle a(|\nabla \psi|)\nabla \psi- a(|\nabla w|)\nabla w, \nabla \varphi \right \rangle {\rm d}x \leq 0, \end{matrix} (\psi-w)^{+}\in W_{0}^{1,B}(B_R), 则在 B_R 上几乎处处有 \psi \leq w. 选取检验函数 \varphi=(\psi-w)^{+}\in W_{0}^{1,B}(B_R) 代入 (2.14) 式, 有 \int_{B_R} \left \langle a(|\nabla \psi|)\nabla \psi- a(|\nabla w|)\nabla w, \nabla ((\psi-w)^{+} )\right \rangle {\rm d}x \leq 0. 于是有 \int_{\{x\in B_R:\psi \geq w\}} \left \langle a(|\nabla \psi|)\nabla \psi- a(|\nabla w|)\nabla w, \nabla ((\psi-w))\right \rangle {\rm d}x \leq 0. 结合引理 2.3(2), 有 \int_{\{x\in B_R:\psi \geq w\}}B(|\nabla \psi-\nabla w|){\rm d}x =0. 这表明在 \{x\in B_R:\psi \geq w\} 上几乎处处有 \nabla \psi=\nabla w, 于是在 B_R 上几乎处处有 \nabla ((\psi-w)^{+} )=0. 又因为 (\psi-w)^{+}\in W_{0}^{1,B}(B_R), 所以在 B_R 上几乎处处有 \psi \leq w. 引理 2.6 得证. 引理2.7 假设 a(|\nabla\psi|)\nabla\psi\in BMO(\Bbb R^{n}), a(|{\boldsymbol f}|){\boldsymbol f}\in BMO(\Bbb R^{n}).$$u$ 是方程(1.1)的 $K_{\psi}^{B}$ - 障碍问题的弱解, $v$ 是边值问题(2.12)的弱解, 则

$\begin{matrix}\label{(2.15)} -\!\!\!\!\!\!\int_{B_R}B(|\nabla u-\nabla v|){\rm d}x \leq C\widetilde{B}\left(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(B_R)}\right)+C\widetilde{B}\left(||a(|\nabla\psi|)\nabla\psi||_{{\rm BMO}(B_R)}\right). \end{matrix}$

$\begin{matrix}\label{(2.16)} {\rm div} (a(|\nabla u|)\nabla u)={\rm div}(a(|{\boldsymbol f}|){\boldsymbol f}-(a(|{\boldsymbol f}|){\boldsymbol f})_{B_{1}}) \end{matrix}$

$K_{\psi}^{B}$ - 障碍问题的弱解. 选取容许函数 $w$, 根据定义 1.1 有

$\begin{eqnarray*}\label{} \int_{\Bbb R^{n}}\left \langle a(|\nabla u|)\nabla u,\nabla(w-u) \right \rangle {\rm d}x \geq \int_{\Bbb R^{n}}\left \langle a(|{\boldsymbol f}|){\boldsymbol f}-(a(|{\boldsymbol f}|){\boldsymbol f})_{B_{1}},\nabla(w-u) \right \rangle {\rm d}x. \end{eqnarray*}$

$\begin{matrix}\label{(2.17)} \int_{B_{1}}\left \langle a(|\nabla u|)\nabla u,\nabla(w-u) \right \rangle {\rm d}x \geq \int_{B_{1}}\left \langle a(|{\boldsymbol f}|){\boldsymbol f}-(a(|{\boldsymbol f}|){\boldsymbol f})_{B_{1}},\nabla(w-u) \right \rangle {\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.27)} &&\int_{B_{1}}\left \langle a(|\nabla w|)\nabla w-a(|\nabla v|)\nabla v,\nabla(w-v) \right \rangle {\rm d}x \nonumber\\ &=& \displaystyle \int_{B_{1}}\left \langle a(|\nabla \psi|)\nabla \psi-(a(|\nabla \psi|)\nabla \psi)_{B_{1}},\nabla(w-v) \right \rangle {\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.28)} && C\int_{B_1}B(|\nabla w-\nabla v|){\rm d}x \nonumber\\ &\leq& \displaystyle \int_{B_{1}}|a(|\nabla \psi|)\nabla \psi-(a(|\nabla \psi|)\nabla \psi)_{B_{1}}||\nabla w-\nabla v| {\rm d}x \nonumber\\ &\leq& \displaystyle \varepsilon_{3} \int_{B_1}B(|\nabla w-\nabla v|){\rm d}x +C(\varepsilon_{3})\int_{B_{1}}\widetilde{B}(a(|\nabla \psi|)\nabla \psi-(a(|\nabla \psi|)\nabla \psi)_{B_{1}}){\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.29)} -\!\!\!\!\!\!\int_{B_1}B(|\nabla w-\nabla v|){\rm d}x \leq C-\!\!\!\!\!\!\int_{B_{1}}\widetilde{B}(a(|\nabla \psi|)\nabla \psi-(a(|\nabla \psi|)\nabla \psi)_{B_{1}}){\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.30)} && -\!\!\!\!\!\!\int_{B_1}B(|\nabla w-\nabla v|){\rm d}x \leq C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{{\rm BMO}(B_{1})}). \end{matrix}$

$\begin{matrix}\label{(2.31)} -\!\!\!\!\!\!\int_{B_1}B(|\nabla u-\nabla v|){\rm d}x &\leq& \displaystyle C-\!\!\!\!\!\!\int_{B_1}B(|\nabla u-\nabla w|){\rm d}x+C-\!\!\!\!\!\!\int_{B_1}B(|\nabla w-\nabla v|){\rm d}x \nonumber\\ &\leq& \displaystyle C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{{\rm BMO}(B_{1})}) + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(B_{1})}). \end{matrix}$

$\begin{matrix}\label{(2.32)} && -\!\!\!\!\!\!\int_{B_{\theta R}(x_0)}B(|\nabla u-(\nabla u)_{B_{\theta R}(x_0)}|){\rm d}x \nonumber\\ &\leq& \displaystyle C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{{\rm BMO}(B_{R}(x_0))}) + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(B_{R}(x_0))}) \nonumber\\ && \displaystyle +\varepsilon_{0}-\!\!\!\!\!\!\int_{B_{R}(x_0)}B(|\nabla u-(\nabla u)_{B_{R}(x_0)}|){\rm d}x \end{matrix}$

$\begin{matrix}\label{(2.33)} -\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-\nabla v|){\rm d}x \leq C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{{\rm BMO}(B_{R})}) + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(B_{R})}). \end{matrix}$

$\begin{matrix}\label{(2.34)} && -\!\!\!\!\!\!\int_{B_{\theta R}}B(|\nabla u-(\nabla u)_{B_{\theta R}}|){\rm d}x \nonumber\\ &\leq& \displaystyle C-\!\!\!\!\!\!\int_{B_{\theta R}}B(|\nabla u-\nabla v|){\rm d}x +C-\!\!\!\!\!\!\int_{B_{\theta R}}B(|(\nabla u)_{B_{\theta R}}-(\nabla v)_{B_{\theta R}}|){\rm d}x \nonumber\\ && \displaystyle +C-\!\!\!\!\!\!\int_{B_{\theta R}}B(|\nabla v-(\nabla v)_{B_{\theta R}}|){\rm d}x \nonumber\\ &:=& \displaystyle K_1+K_2+K_3. \end{matrix}$

$\begin{matrix}\label{(2.35)} K_1=C \theta^{-n}-\!\!\!\!\!\!\int_{B_{ R}}B(|\nabla u-\nabla v|){\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.36)} K_2 &= & \displaystyle C-\!\!\!\!\!\!\int_{B_{\theta R}}B(|-\!\!\!\!\!\!\int_{B_{\theta R}}(\nabla u-\nabla v){\rm d}x|){\rm d}x \nonumber\\ &\leq& \displaystyle C-\!\!\!\!\!\!\int_{B_{\theta R}}B(-\!\!\!\!\!\!\int_{B_{\theta R}}|\nabla u-\nabla v|{\rm d}x){\rm d}x \nonumber\\ &\leq& \displaystyle C-\!\!\!\!\!\!\int_{B_{\theta R}}B(|\nabla u-\nabla v|){\rm d}x \nonumber\\ &\leq& \displaystyle C \theta^{-n}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-\nabla v|){\rm d}x. \end{matrix}$

$\begin{matrix}\label{(2.37)} K_3 &\leq & \displaystyle C \theta^{\sigma}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla v-(\nabla v)_{B_{R}}|){\rm d}x \nonumber\\ &\leq& \displaystyle C\theta^{\sigma}\big( -\!\!\!\!\!\!\int_{B_{ R}}B(|\nabla v-\nabla u|){\rm d}x +-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-(\nabla u)_{B_{R}}|){\rm d}x \nonumber\\ && \displaystyle +-\!\!\!\!\!\!\int_{B_{R}}B(|(\nabla u)_{B_{R}}-(\nabla v)_{B_{R}}|){\rm d}x \big) \nonumber\\ &\leq& \displaystyle C\theta^{\sigma}\big( -\!\!\!\!\!\!\int_{B_{ R}}B(|\nabla v-\nabla u|){\rm d}x +-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-(\nabla u)_{B_{R}}|){\rm d}x \big). \end{matrix}$

$C\theta^{\sigma}=\varepsilon_{0}$, 综合(2.33)-(2.37)式有

$\begin{matrix}\label{(2.38)} && -\!\!\!\!\!\!\int_{B_{\theta R}}B(|\nabla u-(\nabla u)_{B_{\theta R}}|){\rm d}x \nonumber\\ &\leq& \displaystyle C \theta^{-n}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-\nabla v|){\rm d}x +C\theta^{\sigma}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-\nabla v|){\rm d}x \nonumber\\ && \displaystyle +C\theta^{\sigma}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-(\nabla u)_{B_{R}}|){\rm d}x \nonumber\\ &\leq& \displaystyle C \theta^{-n}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-\nabla v|){\rm d}x +C\theta^{\sigma}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-(\nabla u)_{B_{R}}|){\rm d}x \nonumber\\ &\leq& \displaystyle C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{{\rm BMO}(B_{R})}) + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{{\rm BMO}(B_{R})}) \nonumber\\ && \displaystyle +\varepsilon_{0}-\!\!\!\!\!\!\int_{B_{R}}B(|\nabla u-(\nabla u)_{B_{R}}|){\rm d}x. \end{matrix}$

3 主要结论的证明

$\begin{matrix}\label{(3.1)} M_{B_R}^{\sharp}(h,x_{0})= \sup_{0<r<R} -\!\!\!\!\!\!\int _{B_{r}(x_{0})}B(|h-h_{B_{r}(x_{0})}|){\rm d}x. \end{matrix}$

$\begin{matrix}\label{(3.2)} M_{B_R}^{\sharp}(\nabla u,x_{0}) &\leq& C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{BMO(\Bbb R^n)})\nonumber\\ && \displaystyle + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{BMO(\Bbb R^n)}) +\varepsilon_{0} M_{B_R}^{\sharp}(\nabla u,x_{0}). \end{matrix}$

$\begin{matrix}\label{(3.3)} M_{B_R}^{\sharp}(\nabla u,x_{0}) &\leq& C \widetilde{B}(||a(|\nabla \psi|)\nabla \psi||_{BMO(\Bbb R^n)}) + C \widetilde{B}(||a(|{\boldsymbol f}|){\boldsymbol f}||_{BMO(\Bbb R^n)}). \end{matrix}$

$\begin{matrix}\label{(3.4)} B(||\nabla u||_{{\rm BMO}(B_R)})\leq || M_{B_R}^{\sharp}(\nabla u,x_{0})||_{L^{\infty}(B_R)}. \end{matrix}$

$\begin{matrix}\label{(3.5)} B\left(||\nabla u||_{{\rm BMO}(B_R)}\right) \leq \displaystyle C\widetilde{B}\left(||a(|{\boldsymbol f}|){\boldsymbol f}||_{BMO(\Bbb R^{n})}\right)+C\widetilde{B}\left(||a(|\nabla\psi|)\nabla\psi||_{BMO(\Bbb R^{n})}\right). \end{matrix}$

$B\left(||\nabla u||_{BMO(\Bbb R^{n})}\right) \leq \displaystyle C\widetilde{B}\left(||a(|{\boldsymbol f}|){\boldsymbol f}||_{BMO(\Bbb R^{n})}\right)+C\widetilde{B}\left(||a(|\nabla\psi|)\nabla\psi||_{BMO(\Bbb R^{n})}\right).$

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