Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces

2000/09/15 by P. Freitas · 4 citations
Mathematics · #math.SP #math.AP #msc:35P15 #msc:35J05

paper · pdf

published as Indiana Univ. Math. J. 51 (2002), 305--316 · 10 pages

arxiv created 2000/09/15 · arxiv updated 2009/11/30

Abstract

We build a one-parameter family of S¹-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number of S¹-invariant eigenvalues can be made to be arbitrarily small, while the non-invariant spectrum becomes arbitrarily large.

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• Payne's nodal line conjecture fails on doubly-connected planar domains
• Hot spots on cones and warped product manifolds
• Hot spots in domains of constant curvature
• On the nodal domain count under metric perturbations, Courant sharpness and boundary intersections

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