The Dirichlet-to-Neumann operator on rough domains

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

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