The nodal line of the second eigenfunction of the Robin Laplacian in ℝ² can be closed

2010/09/24 by Kennedy, J. B.
#35J05) #35P05 (35B05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.4768

Abstract

We construct a multiply connected domain in ℝ² for which the second eigenfunction of the Laplacian with Robin boundary conditions has an interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains, which does not depend directly on any boundary behaviour. This also gives a new proof of the nodal line property of the example in the Dirichlet case.

Related

• Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions
• On the nodal line of the second eigenfunction of the Laplacian over some concave domains in ℝ²
• Nodal sets of Robin and Neumann eigenfunctions
• Nodal counts for the Robin problem on Lipschitz domains
• Closed nodal surfaces for simply connected domains in higher dimensions