The Dirichlet-to-Neumann operator on rough domains

Abstract

We consider a bounded connected open set $\Omega \subset {\mathrm{R}}^d$ 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.