Sharp estimates for solutions to elliptic problems with mixed boundary conditions

TL;DR

This paper develops sharp estimates for solutions to elliptic PDEs with mixed boundary conditions using symmetrization, enabling comparison between solutions on complex domains and symmetric balls with the same volume.

Contribution

It introduces a symmetrization-based comparison principle for elliptic problems with mixed boundary conditions, extending classical results to more general boundary setups.

Findings

Establishes $L^1$ comparison principles for elliptic solutions

Provides bounds for solutions on irregular domains using symmetric balls

Includes mixed Dirichlet-Robin boundary conditions in the analysis

Abstract

We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $\Omega$ with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as $\Omega$. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.