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 $\phi \mapsto {\partial }_{\nu }u$ 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 & =λu \ \ \text{on}\ Ω,\\ u|_{\partialΩ} & =φ. \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.

16 pages, to appear in Semigroup Forum