Sums of Laplace eigenvalues-rotationally symmetric maximizers in the plane

R. S. Laugesen, B. A. Siudeja

Research output: Contribution to journalArticlepeer-review


The sum of the first n≤1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio (area)3/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.

Original languageEnglish (US)
Pages (from-to)1795-1823
Number of pages29
JournalJournal of Functional Analysis
Issue number6
StatePublished - Mar 15 2011


  • Isoperimetric
  • Membrane
  • Tight frame

ASJC Scopus subject areas

  • Analysis

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