Shape optimization for an elliptic operator with infinitely many positive and negative eigenvalues

TL;DR

This paper investigates shape optimization for elliptic operators with infinitely many eigenvalues, deriving inequalities for extremal eigenvalues and exploring whether classical isoperimetric inequalities apply in this complex setting.

Contribution

It introduces new inequalities for the smallest positive and largest negative eigenvalues and examines the validity of classical isoperimetric inequalities for such operators.

Findings

Derived inequalities for extremal eigenvalues.

Analyzed the applicability of classical isoperimetric inequalities.

Utilized harmonic transplantation and shape derivatives in the analysis.

Abstract

The paper deals with an eigenvalue problems possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domain.