Steklov eigenvalues and quasiconformal maps of simply connected planar domains

TL;DR

This paper establishes sharp upper bounds for sums of Steklov eigenvalues of simply connected planar domains using quasiconformal maps, linking spectral properties to geometric shape deviations.

Contribution

It introduces new isoperimetric bounds for Steklov eigenvalues based on quasiconformal mappings, extending previous spectral geometry results.

Findings

Sharp upper bounds for Steklov eigenvalue sums

Bounds depend on domain's deviation from roundness

Applicable to various spectral functionals including zeta function and heat trace

Abstract

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.