## 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 $$\varphi$$ 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 $$\varphi$$ 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.

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

This paper is available as a pdf (220kB) file. It is also on the arXiv: arxiv.org/abs/1903.05733.

 Tuesday, March 24, 2020