Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

TL;DR

This paper establishes regularity estimates for the polyharmonic Dirichlet problem, highlighting how the positive and negative parts of the right-hand side differently influence the solution, addressing challenges due to the lack of a maximum principle.

Contribution

It introduces a sign-dependent regularity estimate for the polyharmonic Dirichlet problem, providing new insights into the influence of the right-hand side's sign.

Findings

Regularity estimates depend on the sign of the right-hand side

Distinction between positive and negative parts affects solution behavior

Addresses challenges posed by absence of maximum principle

Abstract

A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.