Well-defined spectral position for Neumann domains

Abstract: A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition generated by specific gradient flow lines of the eigenfunction. The restricted eigenfunction onto the partition's components satisfies Neumann boundary conditions and the components are therefore coined 'Neumann domains'. Neumann domains represent a complementary path to the famous nodal-domain partition to study elliptic eigenfunctions where the latter is associated with the Dirichlet Laplacian. A very basic but fundamental property of nodal domains is that the restricted eigenfunction onto a nodal domain always gives the ground-state of the Dirichlet Laplacian. That feature becomes significantly more complex for Neumann domains due to the presence of possible cusps and cracks. In this talk, we focus on this problem and show that the spectral position for Neumann domains is well-defined. Moreover, we provide explicit examples of Neumann domains displaying a fundamentally different behavior in their spectral position than their nodal-domain counterparts.

mathematical physicsanalysis of PDEs

Audience: researchers in the topic

TAMU: Mathematical Physics and Harmonic Analysis Seminar