The Dirichlet-to-Neumann operator on rough domains

The Dirichlet-to-Neumann operator on rough domains

Tue, 14/05
17:00 		Tom ter Elst (Auckland) Functional Analysis Seminar Add to calendar L3
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).