## 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.

: Primary 35J92; secondary 35K92, 35B65, 35B40.

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 Friday, March 21, 2014