论文标题
Weyl的weyl定律在边界粗糙的表面上的steklov问题
Weyl's law for the Steklov problem on surfaces with rough boundary
论文作者
论文摘要
Weyl定律对Lipschitz边界领域的Steklov问题的有效性是光谱几何形状中众所周知的开放问题。我们在二维中回答这个问题，并表明Weyl的定律符合较大的界限，并具有粗糙的边界。该类包括具有内部尖端的域以及“慢”外牙。此外，无法提高外部尖速的状况，这使我们的结果在某种意义上是最佳的。证明是基于萨斯林娜和阿格拉诺维奇的方法，结合了关于保形映射边界行为的一些观察结果。
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.
