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 \(1 < p < \infty\), which acts on the boundary of a bounded Lipschitz domain in \(\mathbb{R}^{d}\) for \(d\ge2\). 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 \((1+\varepsilon)\vee \frac{2d}{d+2}\le p < 2\) for some \(\varepsilon > 0\), 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.