The Dirichlet-to-Neumann operator for divergence form problems

TL;DR

This paper develops an abstract framework for defining and analyzing the Dirichlet-to-Neumann operator in divergence form problems on general Hilbert spaces, especially when classical boundary traces are not well-defined.

Contribution

It introduces a novel abstract approach to Dirichlet-to-Neumann operators, including trace space analogues and a first-order system representation, applicable to complex boundary conditions.

Findings

Defined Dirichlet-to-Neumann operator without classical trace assumptions

Provided a first-order PDE system representation of the operator

Analyzed convergence of operators under coefficient perturbations

Abstract

We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace spaces and are able to give meaning to the Dirichlet-to-Neumann operator of divergence form operators perturbed by a bounded potential in cases where the boundary of the underlying domain does not allow for a well-defined trace. Moreover, a representation of the Dirichlet-to-Neumann operator as a first-order system of partial differential operators is provided. Using this representation, we address convergence of the Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of the leading coefficients converge in the weak operator topology. We also provide some extensions to the case where the bounded potential is not coercive and consider resolvent convergence.