The $p$-Dirichlet-to-Neumann operator

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.