preprint

Inserted: 27 jul 2021

Year: 2021

Abstract:

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 first 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 $\Delta_qu_k=0$. 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. Second, we obtain an asymptotic Courant-type bound for the nodal count of the Dirichlet-to-Neumann eigenfunctions assuming the boundary and the potential are smooth. For the first result, we make use of the duality between the Steklov and Robin problem. To prove the second result we use the variation principle of the Dirichlet-to-Neumann eigenvalues, Weyl asymptotic, and Calderon-Vaillancourt theorem on $L^2$-boundedness of zero order pseudo-differential operators.