Spectral Optimization Problems

TL;DR

This survey explores spectral shape optimization problems involving elliptic PDEs, discussing existence results, open challenges, and related topics like partition problems and evolution flows.

Contribution

It provides a comprehensive overview of spectral shape optimization, highlighting known results and open problems in the field.

Findings

Existence results for spectral optimization problems

Identification of open problems in spectral shape optimization

Connections to related fields like partition problems and flows

Abstract

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.