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arXiv · 2506.14258

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

Simone Ciani·Eurica Henriques·Mariia O. Savchenko·Igor I. Skrypnik
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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.

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Simone Ciani, Eurica Henriques, Mariia O. Savchenko, Igor I. Skrypnik. 2025-06-17. Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions. https://arxiv.org/abs/2506.14258

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