Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

Wolfgang Arendt and Daniel Hauer

Abstract

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\phi$ has a smoothing effect, discovered by Haïm Brezis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $\phi$ are not only closed, but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

Keywords: Nonlinear semigroups, subdifferential, Schaefer's fixed point theorem, existence, smoothing effect, perturbation, compact sublevel sets.

AMS Subject Classification: Primary 35K92; secondary 35K58, 47H20, 47H10.