2022/04/11 by Mikhail Karpukhin, Karpukhin, Mikhail, Jean Lagacé +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2204.05294
openalex publication_date 2022/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.