Minimization of $\lambda_2(\Omega)$ with a perimeter constraint

TL;DR

This paper investigates the problem of minimizing the second Dirichlet eigenvalue of the Laplacian under a perimeter constraint, proving existence and regularity of optimal shapes in two dimensions and more general results in higher dimensions.

Contribution

It establishes the existence, convexity, and boundary regularity of optimal sets in 2D and extends existence results to higher dimensions for a broad class of functionals.

Findings

Optimal sets in 2D are convex with boundary points where curvature vanishes.

Existence of solutions is proven for a class of decreasing, lower semicontinuous functionals in higher dimensions.

Abstract

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.