Nodal sets of Steklov eigenfunctions

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• 作者：Katarína Bellová ; Fang-Hua Lin
• 关键词：Primary 35P99 ; Secondary 35B05 ; 35J05 ; 35S05 ; 47A75
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：2239-2268
• 全文大小：696 KB
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• 作者单位：Katarína Bellová (1)
  Fang-Hua Lin (2)
  
  1. Max Planck Institute for Mathematics in the Sciences, Inselstra?e 22, 04103, Leipzig, Germany
  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in \(\mathbb {R}^n\)- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain \(\Omega \) is \(C^2\), we prove a doubling property for the eigenfunction \(u\). We estimate the Hausdorff \(\mathcal H^{n-2}\)-measure of the nodal set of \(u|_{\partial \Omega }\) in terms of the eigenvalue \(\lambda \) as \(\lambda \) grows to infinity. In case that the domain \(\Omega \) is analytic, we prove a polynomial bound O(\(\lambda ^6\)). Our arguments, which make heavy use of Almgren’s frequency functions, are built on the previous works [Garofalo and Lin, Commun Pure Appl Math 40(3):347-66, 1987; Lin, Commun Pure Appl Math 44(3):287-08, 1991]. Mathematics Subject Classification Primary 35P99 Secondary 35B05 35J05 35S05 47A75