Dirichlet-to-Neumann semigroup with respect to a general second order eigenvalue problem

TL;DR

This paper studies the Dirichlet-to-Neumann operator related to a general second order elliptic PDE, analyzing its semigroup properties such as positivity, sub-Markovianity, irreducibility, and domination under various conditions.

Contribution

It introduces a framework for analyzing the semigroup generated by the Dirichlet-to-Neumann operator for elliptic PDEs with measurable coefficients, extending previous results to more general operators.

Findings

The semigroup exhibits positivity and sub-Markovianity under certain conditions.

Irreducibility and domination properties are established for the associated semigroup.

The study provides foundational results for further analysis of elliptic boundary value problems.

Abstract

In this paper we present a preliminary study on the Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on $L^2(\partial\Omega)$ given by $\varphi\mapsto \partial_{\nu}u$ where $u$ is a weak solution of \begin{equation} \left\{ \begin{aligned} -{\rm div}\, (a\nabla u) +b\cdot \nabla u -{\rm div}\, (cu)+du & =\lambda u \ \ \text{on}\ \Omega,\\ u|_{\partial\Omega} & =\varphi . \end{aligned} \right. \end{equation} Under suitable assumptions on the matrix-valued function $a$, on the vector fields $b$ and $c$, and on the function $d$, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated semigroups.