The p-Dirichlet-to-Neumann operator

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 ${ℝ}^d$, 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.