Karpukhin Mikhail, Lagacé Jean, Polterovich Iosif
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.
Department of Mathematics, King's College London, The Strand, London, WC2R 2LS UK.
Arch Ration Mech Anal. 2023;247(5):77. doi: 10.1007/s00205-023-01912-6. Epub 2023 Aug 10.
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.
对于具有Lipschitz边界的区域上的斯捷克洛夫问题，魏尔定律的有效性是谱几何中一个著名的开放问题。我们在二维情况下回答了这个问题，并表明魏尔定律对于一类边界更粗糙的曲面也成立。这类曲面包括具有内部尖点以及“缓慢”外部尖点的区域。此外，外部尖点速度的条件无法改进，这使得我们的结果在某种意义上是最优的。证明基于苏斯林娜和阿格拉诺维奇的方法，并结合了一些关于共形映射边界行为的观察结果。
