A short note on Meyers' theorem

Abstract

We consider weak solutions of the second-order elliptic equation $\operatorname{div} (\mathbf{M}\nabla u) = \operatorname{div}\mathbf{F}$ in $\Omega\subset {\mathbb R}^d$ with Dirichlet boundary conditions, where $\mathbf{M}$ is a uniformly elliptic real-valued symmetric matrix $\mathbf{M}:\Omega \rightarrow \mathbb{R}^{d\times d}$ such that $\frac{1}{K} |\xi|^2 \leq\langle \mathbf{M}\xi,\xi \rangle \leq K |\xi|^2, \forall \xi \in \mathbb{R}^d, K > 1$. We prove that $\|\nabla u\|_p \leq C\|\mathbf{F}\|_p$ for any $p\in \left(\frac{26K-16}{13K-3}, \frac{26K-16}{13K-13}\right)$.