Nodal count for Dirichlet-to-Neumann operators with potential
- Resource Type
- Working Paper
- Authors
- Hassannezhad, Asma; Sher, David
- Source
- Subject
- Mathematics - Analysis of PDEs
Mathematics - Spectral Theory
58J50 (Primary) 35P15, 58J40, 58C40 (Secondary)
- Language
We consider Dirichlet-to-Neumann operators associated to $\Delta+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $\phi_k$ to the interior satisfying $(\Delta+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $\Delta$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of the duality between the Steklov and Robin problems.
Comment: v.3, 10 pages. Replacement of paper due to a gap in the proof of the result for Dirichlet-to-Neumann eigenfunctions; the bound for the Steklov eigenfunctions is unaffected