PDE Seminar Abstracts

# The Dirichlet-to-Neumann operator on rough domains

Tom ter Elst
University of Auckland, New Zealand
Thursday 18 August 2011 2-3pm, Carslaw 829 (Access Grid Room)

## Abstract

We consider a bounded connected open set $\Omega \subset {ℝ}^{d}$ whose boundary $\Gamma$ has a finite $\left(d-1\right)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator ${D}_{0}$ on ${L}_{2}\left(\Gamma \right)$ by form methods. The operator $-{D}_{0}$ is self-adjoint and generates a contractive ${C}_{0}$-semigroup $S={\left({S}_{t}\right)}_{t>0}$ on ${L}_{2}\left(\Gamma \right)$. We show that the asymptotic behaviour of ${S}_{t}$ as $t\to \infty$ is related to properties of the trace of functions in ${H}^{1}\left(\Omega \right)$ which $\Omega$ may or may not have.

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