@article{2377781ce51645e29e905fe97961a238,
title = "Weyl's Law for the Steklov Problem on Surfaces with Rough Boundary",
abstract = "The validity of Weyl{\textquoteright}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{\textquoteright}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.",
keywords = "Steklov Problem, Conformal methods, Spectral asymptotics",
author = "Mikhail Karpukhin and Jean Lagac{\'e} and Iosif Polterovich",
note = "Funding Information: The authors would like to thank G. Rozenblum for helpful comments and proposed simplifications of the proofs of Proposition 2.2 and Lemma 3.2. We are also grateful to R. Frank, R. Ponge, E. Shargorodsky and A. Ukhlov for useful discussions. The research of M.K. was partially supported by NSF grant DMS-2104254. The research of J.L. was partially supported by EPSRC award EP/T030577/1. He also thanks the University of Bristol for its hospitality while this paper was written. The research of I.P. was partially supported by NSERC and FRQNT. Funding Information: The authors would like to thank G. Rozenblum for helpful comments and proposed simplifications of the proofs of Proposition and Lemma . We are also grateful to R. Frank, R. Ponge, E. Shargorodsky and A. Ukhlov for useful discussions. The research of M.K. was partially supported by NSF grant DMS-2104254. The research of J.L. was partially supported by EPSRC award EP/T030577/1. He also thanks the University of Bristol for its hospitality while this paper was written. The research of I.P. was partially supported by NSERC and FRQNT. Publisher Copyright: {\textcopyright} 2023, The Author(s).",
year = "2023",
month = aug,
day = "10",
doi = "10.1007/s00205-023-01912-6",
language = "English",
volume = "247",
journal = "ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS",
issn = "0003-9527",
publisher = "Springer New York",
number = "5",
}