Nodal sets of Steklov eigenfunctions

Katarína Bellová, Fang-Hua Lin

Research output: Contribution to journalArticle

Abstract

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ℝ<sup>n</sup>- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain Ω is C<sup>2</sup>, we prove a doubling property for the eigenfunction u. We estimate the Hausdorff H<sup>n-2</sup>-measure of the nodal set of u|<inf>∂Ω</inf> 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(λ<sup>6</sup>). 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 languageEnglish (US)
Pages (from-to)2239-2268
Number of pages30
JournalCalculus of Variations and Partial Differential Equations
Volume54
Issue number2
DOIs
StatePublished - Oct 22 2015

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Eigenvalues and eigenfunctions
Eigenfunctions
Dirichlet-to-Neumann Map
Doubling
Probability density function
Bounded Domain
Infinity
Polynomials
Eigenvalue
Polynomial
Estimate

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Nodal sets of Steklov eigenfunctions. / Bellová, Katarína; Lin, Fang-Hua.

In: Calculus of Variations and Partial Differential Equations, Vol. 54, No. 2, 22.10.2015, p. 2239-2268.

Research output: Contribution to journalArticle

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