Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions

TL;DR

This paper introduces a new class of functions related to doubly nonlinear parabolic operators and proves their local boundedness under minimal integrability assumptions, extending classical results to more general settings.

Contribution

It defines the $\\mathcal{PDG}$ class for functions with unbalanced energy estimates and establishes local boundedness without extra integrability, even in sub-critical cases.

Findings

Members of $\mathcal{PDG}$ are locally bounded under various growth conditions.

New bounds are obtained even for classical $p$-Laplace equations in sub-critical cases.

Quantitative a priori estimates are provided following classical methods.

Abstract

We define a suitable class $\mathcal{PDG}$ of functions bearing unbalanced energy estimates, that are embodied by local weak subsolutions to doubly nonlinear, double-phase, Orlicz-type and fully anisotropic operators. Yet we prove that members of $\mathcal{PDG}$ are locally bounded, under critical, sub-critical and limit growth conditions typical of singular parabolic operators, with quantitative a priori estimates that follow the lines of the pioneering work of Ladyzhenskaya, Solonnikov and Uraltseva \cite{LadSolUra}. These local bounds are new in the sub-critical cases, even for the classic $p$-Laplacean equations, since no extra-integrability condition is needed.