A surgery result for the spectrum of the Dirichlet Laplacian

TL;DR

This paper introduces a geometric modification method for open sets that controls the first k eigenvalues of the Dirichlet Laplacian, perimeter, and diameter, with potential applications in shape optimization.

Contribution

It provides a novel geometric surgery technique to modify shapes while controlling spectral and geometric properties, based on shape subsolutions for torsion energy.

Findings

First k eigenvalues are non-increasing after modification

Perimeter and diameter can be reduced below a threshold

The measure of the set remains constant

Abstract

In this paper we give a method to geometrically modify an open set such that the first $k$ eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy.