Sharp Sobolev regularity for widely degenerate parabolic equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909627519598592 |
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| author | Ambrosio, Pasquale |
| author_facet | Ambrosio, Pasquale |
| contents | We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ Ω_{T}=Ω\times(0,T), \] where $p\geq2$, $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a non-negative constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Besov parabolic space when $p>2$ and that $f\in L_{loc}^{2}(Ω_{T})$ if $p=2$, we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary $p$-Poisson equation. The main novelty here is that $f$ only has a Besov or Lebesgue spatial regularity, unlike the previous work [6], where $f$ was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [5], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_05432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Sobolev regularity for widely degenerate parabolic equations Ambrosio, Pasquale Analysis of PDEs 35B45, 35B65, 35D30, 35K10, 35K65 We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ Ω_{T}=Ω\times(0,T), \] where $p\geq2$, $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a non-negative constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Besov parabolic space when $p>2$ and that $f\in L_{loc}^{2}(Ω_{T})$ if $p=2$, we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary $p$-Poisson equation. The main novelty here is that $f$ only has a Besov or Lebesgue spatial regularity, unlike the previous work [6], where $f$ was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [5], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations. |
| title | Sharp Sobolev regularity for widely degenerate parabolic equations |
| topic | Analysis of PDEs 35B45, 35B65, 35D30, 35K10, 35K65 |
| url | https://arxiv.org/abs/2407.05432 |