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

Abstract

In this paper we study a 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

$$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div}\, (a\nabla u) +b\cdot \nabla u -\mathrm{div}\, (cu)+du =\lambda u \ \ \text {on}\ \Omega ,\\&u|_{\partial \Omega } =\varphi . \end{aligned} \right. \end{aligned}$$

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 Dirichlet-to-Neumann semigroups.