The Dirichlet-to-Neumann operator via hidden compactness

TL;DR

This paper introduces a novel approach to defining and analyzing the Dirichlet-to-Neumann operator for symmetric elliptic operators on Lipschitz domains, using hidden compactness to handle non-coerciveness and establishing convergence results.

Contribution

It develops a new framework employing hidden compactness to define self-adjoint Dirichlet-to-Neumann operators, including multi-valued cases, and proves resolvent and semigroup convergence under coefficient perturbations.

Findings

Established a self-adjoint Dirichlet-to-Neumann operator via hidden compactness.

Proved uniform resolvent convergence for sequences of operators with converging coefficients.

Demonstrated semigroup convergence under the unique continuation property.

Abstract

We show that to each symmetric elliptic operator of the form \[ \mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c \] on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ one can associate a self-adjoint Dirichlet-to-Neumann operator on $L_2(\partial \Omega)$, which may be multi-valued if 0 is in the Dirichlet spectrum of $\mathcal{A}$. To overcome the lack of coerciveness in this case, we employ a new version of the Lax--Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever their coefficients converge uniformly and the second-order limit operator in $L_2(\Omega)$ has the unique continuation property. We also consider semigroup convergence.