A short note on Meyers' theorem

Abstract

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

5 pages