Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

Christopher D. Sogge, Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.

Original languageEnglish (US)
Pages (from-to)1249-1289
Number of pages41
JournalCommunications in Partial Differential Equations
Volume42
Issue number8
DOIs
StatePublished - Aug 3 2017

Keywords

  • Cauchy data
  • Laplacian
  • concave boundary
  • eigenfunction
  • glancing parametrix
  • glancing set

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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