Nodal sets of Steklov eigenfunctions

Katarína Bellová; Fang Hua Lin

doi:10.1007/s00526-015-0864-8

Nodal sets of Steklov eigenfunctions

Katarína Bellová, Fang Hua Lin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ℝⁿ- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain Ω is C², we prove a doubling property for the eigenfunction u. We estimate the Hausdorff H^n-2-measure of the nodal set of u|_∂Ω in terms of the eigenvalue λ as λ grows to infinity. In case that the domain Ω is analytic, we prove a polynomial bound O(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].

Original language	English (US)
Pages (from-to)	2239-2268
Number of pages	30
Journal	Calculus of Variations and Partial Differential Equations
Volume	54
Issue number	2
DOIs	https://doi.org/10.1007/s00526-015-0864-8
State	Published - Oct 22 2015

Keywords

35J05
35S05
47A75
Primary 35P99
Secondary 35B05

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1007/s00526-015-0864-8

Cite this

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title = "Nodal sets of Steklov eigenfunctions",

abstract = "We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ℝn- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain Ω is C2, we prove a doubling property for the eigenfunction u. We estimate the Hausdorff Hn-2-measure of the nodal set of u|∂Ω in terms of the eigenvalue λ as λ grows to infinity. In case that the domain Ω is analytic, we prove a polynomial bound O(λ6). Our arguments, which make heavy use of Almgren{\textquoteright}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].",

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AB - We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ℝn- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain Ω is C2, we prove a doubling property for the eigenfunction u. We estimate the Hausdorff Hn-2-measure of the nodal set of u|∂Ω in terms of the eigenvalue λ as λ grows to infinity. In case that the domain Ω is analytic, we prove a polynomial bound O(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].

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Nodal sets of Steklov eigenfunctions

Abstract

Keywords

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