Pointwise Inequalities for Elliptic Boundary Value Problems

TL;DR

This paper presents a novel approach to deriving pointwise estimates for elliptic boundary value problem solutions using weighted integral inequalities, demonstrated on various elliptic equations and systems.

Contribution

Introduces a new method leveraging weighted integral inequalities to obtain pointwise estimates for elliptic boundary value problems, applicable to multiple equations and systems.

Findings

Method successfully applied to scalar second-order elliptic equations

Extended to 3D Lamé system and higher-order elliptic equations

Techniques can be adapted to other elliptic problems with similar inequalities

Abstract

We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lam\'{e} system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied.