Weyl's law for the Steklov problem on surfaces with rough boundary

TL;DR

This paper proves Weyl's law for the Steklov problem on two-dimensional surfaces with very rough boundaries, including cusps, extending previous results and demonstrating optimal conditions for boundary irregularities.

Contribution

It establishes Weyl's law validity for a broad class of rough-boundary surfaces, including those with interior and exterior cusps, using advanced boundary analysis techniques.

Findings

Weyl's law holds for surfaces with rough boundaries in two dimensions.

The class of domains includes those with interior and slow exterior cusps.

The boundary conditions for the law's validity are shown to be optimal.

Abstract

The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundaries is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as 'slow' exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.