The Dirichlet-to-Neumann operator on rough domains

Tom ter Elst
University of Auckland, NZ
Thu 18 August 2010 2-3pm, Carslaw 829 (Access Grid Room), note the unusual day.

Abstract

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

The talk is based on joint work with W. Arendt (Ulm).

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.