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Weyl's Law for the Steklov Problem on Surfaces with Rough Boundary
Mikhail Karpukhin1, Jean Lagacé2, Iosif Polterovich3
1Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.
Archive for Rational Mechanics and Analysis
|August 14, 2023
Summary
Weyl's law validity for the Steklov problem is confirmed for domains with Lipschitz boundaries in 2D. This extends to surfaces with interior and exterior cusps, improving understanding in spectral geometry.
Area of Science:
- Spectral geometry
- Mathematical analysis
Background:
- Weyl's law for the Steklov problem on domains with Lipschitz boundary is an open question.
- Understanding spectral properties of domains with irregular boundaries is crucial.
Purpose of the Study:
- To determine the validity of Weyl's law for the Steklov problem on domains with Lipschitz boundary in two dimensions.
- To extend the validity of Weyl's law to a broader class of surfaces with rough boundaries.
Main Methods:
- Application of spectral geometry principles.
- Utilizing methods developed by Suslina and Agranovich.
- Analysis of boundary behavior of conformal mappings.
Main Results:
- Weyl's law is proven to be valid for the Steklov problem on 2D domains with Lipschitz boundary.
- The law's validity is extended to surfaces featuring interior cusps and 'slow' exterior cusps.
- The condition for exterior cusps is shown to be optimal, indicating the result's significance.
Conclusions:
- The study resolves a long-standing open question in spectral geometry.
- The findings broaden the applicability of Weyl's law to more complex domains.
- The optimality of the results provides a benchmark for future research.
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