The boundary Harnack principle on optimal domains

TL;DR

This paper provides a concise proof of the Boundary Harnack inequality for specific domains with geometric conditions, and explores applications in shape optimization and free boundary problems.

Contribution

It introduces a new, self-contained proof of the Boundary Harnack principle for domains characterized by a state function with particular properties.

Findings

Proved Boundary Harnack inequality under geometric conditions.

Established applications to shape optimization problems.

Discussed implications for free boundary problems.

Abstract

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.