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Nodal Sets of Laplace Eigenfunctions: Estimates of the Hausdorff Measure in Dimensions Two and Three

  • Alexander Logunov
  • Eugenia Malinnikova
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 261)

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.

Keywords

Laplace eigenfunctions nodal set harmonic functions 

Mathematics Subject Classification (2010)

Primary 31B05 Secondary 35R01 58G25 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Chebyshev Lab.St. Petersburg State UniversitySt. PetersburgRussia
  3. 3.Dept. of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  4. 4.Dept. of MathematicsPurdue UniversityWest LafayetteUSA

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