PreprintThe \(p\)DirichlettoNeumann operator with applications to elliptic and parabolic problemsDaniel HauerAbstractIn this paper, we investigate the DirichlettoNeumann operator associated with second order quasilinear 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 wellposedness and Höldercontinuity with uniform estimates of weak solutions of some elliptic boundaryvalue problems involving the DirichlettoNeumann operator. By employing these regularity results of weak solutions of elliptic problems, we show that the semigroup generated by the negative DirichlettoNeumann operator on \(L^{q}\) enjoys an \(L^{q}C^{0,\alpha}\)smoothing effect and the negative DirichlettoNeumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and orderpreserving 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.
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