PreprintMaximal \(L^2\)regularity in nonlinear gradient systems and perturbations of sublinear growthWolfgang Arendt and Daniel HauerAbstractThe 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 Nemytskiioperator 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 DirichlettoNeumann 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.
This paper is available as a pdf (220kB) file. It is also on the arXiv: arxiv.org/abs/1903.05733.
