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

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.