The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set

$\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite

$(d-1)$-dimensional Hausdorff measure. Then we define the

Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form

methods. The operator $-D_0$ is self-adjoint and generates a

contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on

$L_2(\Gamma)$. We show that the asymptotic behaviour of

$S_t$ as $t \to \infty$ is related to properties of the

trace of functions in $H^1(\Omega)$ which $\Omega$ may or

may not have. We also show that they are related to the

essential spectrum of the Dirichlet-to-Neumann operator.

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