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

## TL;DR

This paper establishes maximal $L^2$-regularity for nonlinear gradient systems with sublinear perturbations, extending classical smoothing effects and applying fixed point methods to solve perturbed evolution equations.

## Contribution

It introduces a novel approach combining smoothing effects and fixed point theorems to handle sublinear perturbations in nonlinear gradient systems.

## Key findings

- Maximal $L^2$-regularity holds for the perturbed evolution equations.
- The method applies to the $p$-Laplacian and $p$-harmonic Dirichlet-to-Neumann operators.
- Sublevel sets' compactness is crucial for the results.

## Abstract

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\varphi$ has a smoothing effect, discovered by H. Br\'ezis, 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.