Weyl's Law for the Steklov Problem on Surfaces with Rough Boundary
Related Experiment Video
Updated: Jul 19, 2025

Determination of Aggregate Surface Morphology at the Interfacial Transition Zone ITZ
Published on: December 16, 2019
Mikhail Karpukhin1, Jean Lagacé2, Iosif Polterovich3
1Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.
Abstract:
The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundary 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.
More Related Videos
11:47Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments
Published on: February 27, 2013
00:07Accurate Determination of the Equilibrium Surface Tension Values with Area Perturbation Tests
Published on: August 30, 2019
Related Concept Videos
Divergence and Stokes' Theorems
Electrostatic Boundary Conditions
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
Generalized Hooke's Law
Bending of Curved Members - Strain Analysis
The important part of bending analysis for such a member...
Bending of Curved Members - Neutral Surface
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within...
Equipotential Surfaces and Field Lines