The $p$-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems

Daniel Hauer

Abstract

In this paper, we investigate the Dirichlet-to-Neumann operator associated with second order quasi-linear operators of $p$-Laplace type for $1<p<\mathrm{\infty }$, which acts on the boundary of a bounded Lipschitz domain in ${\mathbb{R}}^d$ for $d\ge 2$. We establish well-posedness and Hölder-continuity with uniform estimates of weak solutions of some elliptic boundary-value problems involving the Dirichlet-to-Neumann operator. By employing these regularity results of weak solutions of elliptic problems, we show that the semigroup generated by the negative Dirichlet-to-Neumann operator on $L^q$ enjoys an $L^q-C^{0,\alpha }$-smoothing effect and the negative Dirichlet-to-Neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order-preserving semigroup. Moreover, we establish convergence in large time with decay rates of all trajectories of the semigroup, and in the singular case $(1+\epsilon )\vee \frac{2d}{d+2}\le p<2$ for some $\epsilon >0$, we give upper estimates of the finite time of extinction.

Keywords: Elliptic problems, Parabolic problems, Hölder regularity, Nonlocal operator, $p$-Laplace operator, Asymptotic behaviour.

AMS Subject Classification: Primary 35J92; secondary 35K92, 35B65, 35B40.