Nodal sets of laplace eigenfunctions: Estimates of the hausdorff measure in dimensions two and three

Alexander Logunov, Eugenia Malinnikova

Research output: Chapter in Book/Report/Conference proceedingChapter

22 Scopus citations

Abstract

Let ΔM be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u: ΔMu+λu = 0. In dimension n = 2 we refine the Donnelly–Fefferman estimate by showing that H1({u = 0}) ≤Cλ3/4−β for some β∈ (0, 1/4). The proof employs the Donnelly–Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension n = 3: H2({u = 0}) ≥ cλα for some α∈ (0,1/2). The positive constants c, C depend on the manifold, α and β are universal.

Original languageEnglish (US)
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Pages333-344
Number of pages12
DOIs
StatePublished - 2018

Publication series

NameOperator Theory: Advances and Applications
Volume261
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Harmonic functions
  • Laplace eigenfunctions
  • Nodal set

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