Gradient higher integrability for degenerate/ singular parabolic multi-phase problems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916477109534720 |
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| author | Sen, Abhrojyoti |
| author_facet | Sen, Abhrojyoti |
| contents | This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of $p$-Laplace type is given by \[ u_t - \operatorname{div} \left(|\nabla u|^{p-2} \nabla u + a(z) |\nabla u|^{q-2} \nabla u + b(z) |\nabla u|^{s-2} \nabla u \right) = 0, \] where $\frac{2n}{n+2} < p \leq q \leq s < \infty$, and the coefficients $a(z)$ and $b(z)$ are non-negative Hölder continuous functions on $Ω_T = Ω\times (0, T)$, with $Ω\subset \mathbb{R}^n$. We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ($p \geq 2$) and the singular regime $\left(\frac{2n}{n+2} < p < 2\right),$ providing a unified framework. Our approach involves proving uniform parabolic Sobolev-Poincaré inequalities, which are key to establishing reverse Hölder type inequalities, along with covering lemmas for the $p$, $(p,q)$, $(p,s)$, and $(p,q,s)$-phases without distinguishing between the regimes of $p$, $q$, and $s$. In the end, we also discuss the gradient higher integrability for general parabolic multi-phase problem involving a finite number of phases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00763 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient higher integrability for degenerate/ singular parabolic multi-phase problems Sen, Abhrojyoti Analysis of PDEs This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of $p$-Laplace type is given by \[ u_t - \operatorname{div} \left(|\nabla u|^{p-2} \nabla u + a(z) |\nabla u|^{q-2} \nabla u + b(z) |\nabla u|^{s-2} \nabla u \right) = 0, \] where $\frac{2n}{n+2} < p \leq q \leq s < \infty$, and the coefficients $a(z)$ and $b(z)$ are non-negative Hölder continuous functions on $Ω_T = Ω\times (0, T)$, with $Ω\subset \mathbb{R}^n$. We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ($p \geq 2$) and the singular regime $\left(\frac{2n}{n+2} < p < 2\right),$ providing a unified framework. Our approach involves proving uniform parabolic Sobolev-Poincaré inequalities, which are key to establishing reverse Hölder type inequalities, along with covering lemmas for the $p$, $(p,q)$, $(p,s)$, and $(p,q,s)$-phases without distinguishing between the regimes of $p$, $q$, and $s$. In the end, we also discuss the gradient higher integrability for general parabolic multi-phase problem involving a finite number of phases. |
| title | Gradient higher integrability for degenerate/ singular parabolic multi-phase problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.00763 |