**SMS scnews item created by Daniel Daners at Mon 17 Mar 2014 1135**

Type: Seminar

Distribution: World

Expiry: 17 Mar 2014

**Calendar1: 17 Mar 2014 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@como.maths.usyd.edu.au

### PDE Seminar

# The p-Dirichlet-to-Neumann operator

### Hauer

Daniel Hauer

University of Sydney

17 March 2014 14:00-15:00, Carslaw Room 829 (AGR)

## Abstract

In this talk we are interested in the Dirichlet-to-Neumann operator associated with
the $p$-Laplace
operator on a bounded Lipschitz domain in
${\mathbb{R}}^{d}$, where
$1<p<\infty $ and
$d\ge 2$. If
$p\ne 2$,
then the Dirichlet-to-Neumann operator becomes nonlinear and not much
was known so far. We outline how one obtains well-posedness and
Hölder-regularity of weak solutions of some elliptic problems associated with the
Dirichlet-to-Neumann operator. Further, we show that the semigroup generated
by the negative Dirichlet-to-Neumann operator can be extrapolated on all
${L}^{q}$-spaces and enjoys an
interesting ${L}^{q}-{C}^{0,\alpha}$-smoothing
effect. Moreover, we outline how the part of the Dirichlet-to-Neumann
operator in the space of continuous functions on the boundary is
$m$-accretive
and give a sufficient condition to ensure that the negative operator generates a
strongly continuous semigroup on this space. We conclude this talk by stating
some results to the large time stability of the semigroup and give decay rates.

Check also the PDE
Seminar page. Enquiries to Daniel Hauer or Daniel Daners.