Nodal count for Dirichlet-to-Neumann operators with potential

2021/07/07 by Hassannezhad, Asma, Sher, David · 1 citation
#58C40 (Secondary) #58J40 #58J50 (Primary) 35P15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2107.03370

Abstract

We consider Dirichlet-to-Neumann operators associated to Δ+q on a Lipschitz domain in a smooth manifold, where q is an L^∞ potential. We prove a Courant-type bound for the nodal count of the extensions u_k of the kth Dirichlet-to-Neumann eigenfunctions ϕ_k to the interior satisfying (Δ+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 Δ. 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.

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• Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
• Nodal domains a la Courant
• Nodal Sets of Steklov Eigenfunctions
• Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions
• A lower bound for the nodal sets of Steklov eigenfunctions