Nodal sets of Steklov eigenfunctions
Nodal sets of Steklov eigenfunctions
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Steklov 特征函数的节点集
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
F. Lin
中科院分区：
文献类型：
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作者：
K. Bellová;F. Lin
We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $$\mathbb {R}^n$$Rn- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain $$\Omega $$Ω is $$C^2$$C2, we prove a doubling property for the eigenfunction $$u$$u. We estimate the Hausdorff $$\mathcal H^{n-2}$$Hn-2-measure of the nodal set of $$u|_{\partial \Omega }$$u|∂Ω in terms of the eigenvalue $$\lambda $$λ as $$\lambda $$λ grows to infinity. In case that the domain $$\Omega $$Ω is analytic, we prove a polynomial bound O($$\lambda ^6$$λ6). 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):347–366, 1987; Lin, Commun Pure Appl Math 44(3):287–308, 1991].
