Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems

TL;DR

This paper investigates how Steklov-type eigenvalues related to Sobolev trace constants change with domain shape, deriving formulas for shape derivatives and identifying critical domains like balls under volume or perimeter constraints.

Contribution

It establishes real-analytic dependence of elementary symmetric functions of eigenvalues on domain variations and characterizes critical domains via overdetermined systems.

Findings

Elementary symmetric functions of eigenvalues depend real-analytically on domain shape.

Derived Hadamard-type formulas for shape derivatives of eigenvalues.

Balls are identified as critical domains under volume or perimeter constraints.

Abstract

We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the elementary symmetric functions of the eigenvalues subject to volume or perimeter constraint.