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

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.