Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915348295450624 |
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| author | Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. |
| author_facet | Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. Analysis of PDEs 35B65, 35B45, 35K65 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. |
| title | Parabolic De Giorgi classes with doubly nonlinear, nonstandard growth: local boundedness under exact integrability assumptions |
| topic | Analysis of PDEs 35B65, 35B45, 35K65 |
| url | https://arxiv.org/abs/2506.14258 |