The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D0 on L2(Γ) by form methods. The operator −D0 is self-adjoint and generates a contractive C0-semigroup S=(St)t>0 on L2(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H1(Ω) which Ω may or may not have.