Weighted a priori estimates for the solution of the homogeneous Dirichlet problem for the powers of the Laplacian Operator

Abstract

Let $u$ be a weak solution of $(-\Delta {)}^{m}u=f$ with Dirichlet boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$. Then, the main goal of this paper is to prove the following a priori estimate: $$\Vert u{\Vert }_{W_{\omega }^{2m,p}(\Omega )}\le C\Vert f{\Vert }_{L_{\omega }^p(\Omega )},$$ where $\omega$ is a weight in the Muckenhoupt class $A_p.$