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 Cîrstea or Daniel Daners.