Sharp spectral bounds on starlike domains

Richard S. Laugesen, Bartłomiej A. Siudeja

Research output: Contribution to journalArticlepeer-review

Abstract

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber-Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by Pólya and Szego. Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.

Original languageEnglish (US)
Pages (from-to)309-347
Number of pages39
JournalJournal of Spectral Theory
Volume4
Issue number2
DOIs
StatePublished - 2014

Keywords

  • Heat trace
  • Isoperimetric
  • Membrane
  • Partition function
  • Sloshing
  • Spectral zeta

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Geometry and Topology

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