Shape flows for spectral optimization problems

TL;DR

This paper develops a general framework for gradient flow evolution in metric spaces, focusing on capacitary measures and domain evolution in spectral optimization, providing new insights into their dynamic behavior.

Contribution

It introduces a novel approach to analyze gradient flows for spectral optimization problems within metric space settings, extending existing methods to capacitary measures and domain evolution.

Findings

Established a gradient flow framework for capacitary measures

Analyzed domain evolution in spectral optimization problems

Provided new tools for metric space-based spectral analysis

Abstract

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the $\gamma$-convergence dissipation distance and with the evolution of domains in spectral optimization problems.