Prescription du spectre de Steklov dans une classe conforme

TL;DR

This paper demonstrates the ability to prescribe parts of the Steklov spectrum within a conformal class on compact manifolds, and establishes bounds on eigenvalue multiplicities for various cases.

Contribution

It shows how to prescribe finite parts of the Steklov spectrum in a conformal class and provides bounds on eigenvalue multiplicities on surfaces and disks.

Findings

Any finite part of the Steklov spectrum can be prescribed within a conformal class.

The multiplicity of the first eigenvalue on a surface is unbounded, but bounded on a disk.

For the Steklov-Neumann problem on the disk, the eigenvalue multiplicity is at most k+1.

Abstract

On any compact manifold of dimension $n\geq3$ with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the $k$-th eigenvalue is bounded independently of the metric. On the disk, we give more precise results : the multiplicity of the first and second positive eigenvalues are at most 2 and 3 respectively. For the Steklov-Neumann problem on the disk, we prove that the multiplicity of the $k$-th positive eigenvalue is at most $k+1$.