### An approximating family for the Dirichlet-to-Neumann semigroup

#### Abstract

In this paper we prove that the Dirichlet-to-Neumann semigroup $S(t)$ is an analytic compact Markov irreducible semigroup in $C(\partial \Omega)$ in any bounded smooth domain $\Omega$. By a generalization of the Lax semigroup, we construct an approximating family for $S(t)$. We prove some regularizing characters and compactness of this family. By using the ergodic properties of $S(t)$, we deduce its asymptotic behavior. At the end we conjecture some open problems.

#### Article information

Source
Adv. Differential Equations Volume 11, Number 3 (2006), 241-257.

Dates
First available in Project Euclid: 18 December 2012