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

TL;DR

This paper establishes weighted a priori estimates for solutions to the homogeneous Dirichlet problem involving powers of the Laplacian, extending regularity results to weighted Sobolev spaces with Muckenhoupt weights.

Contribution

It proves new weighted a priori estimates for higher-order Laplacian problems in bounded domains with Muckenhoupt weights.

Findings

Weighted estimates hold in $W^{2m,p}_ heta(\

 ext{ with weights in }A_p$

Extension of regularity theory to weighted Sobolev spaces for higher-order Laplacian problems.

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: $$ \|u\|_{W^{2m,p}_\omega(\Omega)} \le C\, \|f\|_{L^p_\omega(\Omega)}, $$ where $\omega$ is a weight in the Muckenhoupt class $A_p.$