The Dirichlet-to-Neumann operator on rough domains

Abstract

We consider a bounded connected open set $\Omega \subset {ℝ}^d$ whose boundary $\Gamma$ has a finite $\left(d-1\right)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2\left(\Gamma \right)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S={\left(S_t\right)}_{t>0}$ on $L_2\left(\Gamma \right)$. We show that the asymptotic behaviour of $S_t$ as $t\to \infty$ is related to properties of the trace of functions in $H^1\left(\Omega \right)$ which $\Omega$ may or may not have.

The talk is based on joint work with W. Arendt (Ulm).