Boundary Estimates for Doubly Nonlinear Parabolic Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913402481278976 |
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| author | Gianazza, Ugo Jesus, David |
| author_facet | Gianazza, Ugo Jesus, David |
| contents | We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder $Ω_T$, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain $Ω_T$, we obtain a power-like decay at the boundary and a boundary Harnack inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary Estimates for Doubly Nonlinear Parabolic Equations Gianazza, Ugo Jesus, David Analysis of PDEs 35K92, 35K67, 35B65 (Primary) 35B45 (Secondary) We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder $Ω_T$, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain $Ω_T$, we obtain a power-like decay at the boundary and a boundary Harnack inequality. |
| title | Boundary Estimates for Doubly Nonlinear Parabolic Equations |
| topic | Analysis of PDEs 35K92, 35K67, 35B65 (Primary) 35B45 (Secondary) |
| url | https://arxiv.org/abs/2406.16096 |