Some New Problems in Spectral Optimization

TL;DR

This paper introduces new spectral optimization problems involving domain shape, graph structure, and Schrödinger potentials across various geometric and metric spaces, expanding the scope of spectral analysis.

Contribution

It formulates novel spectral optimization problems in diverse settings including Riemannian, Finsler, Carnot-Carathéodory, Gaussian spaces, graphs, and Schrödinger potentials.

Findings

New formulations of spectral optimization problems.

Application to diverse geometric and metric spaces.

Potential for new analytical and computational methods.

Abstract

We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the {\it metric Laplacian}, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carath\'eodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schr\"odinger potential in suitable classes.