Abstract:
We consider a bounded connected open set Ω⊂RdΩ⊂Rd whose boundary Γ has a finite (d−1)(d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D0D0 on L2(Γ)L2(Γ) by form methods. The operator −D0−D0 is self-adjoint and generates a contractive C0C0-semigroup S=(St)t>0S=(St)t>0 on L2(Γ)L2(Γ). We show that the asymptotic behaviour of StSt as t→∞t→∞ is related to properties of the trace of functions in H1(Ω)H1(Ω) which Ω may or may not have.