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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.03834 |
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| _version_ | 1866909129721774080 |
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| author | Wang, Yuqing Zhu, Yizhe |
| author_facet | Wang, Yuqing Zhu, Yizhe |
| contents | Denote by $Δ$ the Laplacian and by $Δ_\infty$ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^{\infty}$, $$\bigg| |D^2vDv|^2-ΔvΔ_\infty v-\frac{1}{2}[|D^2v|^2-(Δv)^2]|Dv|^2\bigg| \le\frac{n-2}{2}[|D^2v|^2|Dv|^2-|D^2vDv|^2]$$ Based on this, we prove the result: When $n\ge2$ and $p(x)\in(1,2)\cup(2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the $W^{2,2}_{loc}$-regularity in the spatial variable and the $W^{1,2}_{loc}$-regularity in the time variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure Wang, Yuqing Zhu, Yizhe Analysis of PDEs Denote by $Δ$ the Laplacian and by $Δ_\infty$ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^{\infty}$, $$\bigg| |D^2vDv|^2-ΔvΔ_\infty v-\frac{1}{2}[|D^2v|^2-(Δv)^2]|Dv|^2\bigg| \le\frac{n-2}{2}[|D^2v|^2|Dv|^2-|D^2vDv|^2]$$ Based on this, we prove the result: When $n\ge2$ and $p(x)\in(1,2)\cup(2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the $W^{2,2}_{loc}$-regularity in the spatial variable and the $W^{1,2}_{loc}$-regularity in the time variable. |
| title | Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.03834 |