An overview on constrained critical points of Dirichlet integrals

TL;DR

This paper explores the critical points of Dirichlet integrals under $L^q$ constraints, generalizing eigenvalue problems for the Laplacian, and discusses results, counterexamples, and open questions.

Contribution

It provides a comprehensive overview of constrained critical points of Dirichlet integrals, including new results, counterexamples, and open problems in the field.

Findings

Collected various results on constrained critical points.

Presented counterexamples challenging existing assumptions.

Outlined open problems for future research.

Abstract

We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit $L^q$ sphere. We collect some results, present some counter-examples and compile a list of open problems.