Vol. 2, No. 1, 2020

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

Wolfgang Arendt and Daniel Hauer

Vol. 2 (2020), No. 1, 23–34
DOI: 10.2140/paa.2020.2.23
Abstract

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function φ 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 φ 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
Mathematical Subject Classification 2010
Primary: 35K92, 35K58, 47H20, 47H10
Milestones
Received: 14 February 2019
Revised: 3 August 2019
Accepted: 9 September 2019
Published: 9 November 2019
Authors
Wolfgang Arendt
Institute of Applied Analysis
University of Ulm
Ulm
Germany
Daniel Hauer
School of Mathematics and Statistics
The University of Sydney
Sydney
Australia