Regularity theory for parabolic operators in the half-space with boundary degeneracy
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914797570752512 |
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| author | Metafune, Giorgio Negro, Luigi Spina, Chiara |
| author_facet | Metafune, Giorgio Negro, Luigi Spina, Chiara |
| contents | We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*}
\mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+Cy^{α_2-1}D_y \end{align*} under Neumann boundary condition, in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$. We prove elliptic and parabolic $L^p$-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that $\mathcal L$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_14319 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity theory for parabolic operators in the half-space with boundary degeneracy Metafune, Giorgio Negro, Luigi Spina, Chiara Analysis of PDEs 35K67, 35B45, 47D07, 35J70, 35J75 We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+Cy^{α_2-1}D_y \end{align*} under Neumann boundary condition, in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$. We prove elliptic and parabolic $L^p$-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that $\mathcal L$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity. |
| title | Regularity theory for parabolic operators in the half-space with boundary degeneracy |
| topic | Analysis of PDEs 35K67, 35B45, 47D07, 35J70, 35J75 |
| url | https://arxiv.org/abs/2309.14319 |