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 \(\Omega \subset \mathbb R^d\) 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.

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.