Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910448555655168 |
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| author | Negro, Luigi |
| author_facet | Negro, Luigi |
| contents | We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+y^{\frac{α_1+α_2}{2}-1}\left(d,\nabla_x\right)+cy^{α_2-1}D_y-by^{α_2-2}$$ in the half-space $\mathcal{R}^{N+1}_+=\{(x,y): x \in \mathcal{R}^N, y>0\}$, under Dirichlet or oblique derivative boundary conditions. In the special case $α_1=α_2=α$ the operator $\mathcal L$ takes the form $$ \mathcal L=y^α\mbox{Tr }\left(AD^2\right)+y^{α-1}\left(v,\nabla\right)-by^{α-2},$$ where $v=(d,c)\in\mathcal{R}^{N+1}$, $b\in\mathcal{R}$ and $ A=\left( \begin{array}{c|c}
Q & { q}^t \\[1ex] \hline
q& γ\end{array}\right)$ is an elliptic matrix. 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 |
| id |
arxiv_https___arxiv_org_abs_2405_09540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions Negro, Luigi Analysis of PDEs 35K67, 35B45, 47D07, 35J70, 35J75 We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+y^{\frac{α_1+α_2}{2}-1}\left(d,\nabla_x\right)+cy^{α_2-1}D_y-by^{α_2-2}$$ in the half-space $\mathcal{R}^{N+1}_+=\{(x,y): x \in \mathcal{R}^N, y>0\}$, under Dirichlet or oblique derivative boundary conditions. In the special case $α_1=α_2=α$ the operator $\mathcal L$ takes the form $$ \mathcal L=y^α\mbox{Tr }\left(AD^2\right)+y^{α-1}\left(v,\nabla\right)-by^{α-2},$$ where $v=(d,c)\in\mathcal{R}^{N+1}$, $b\in\mathcal{R}$ and $ A=\left( \begin{array}{c|c} Q & { q}^t \\[1ex] \hline q& γ\end{array}\right)$ is an elliptic matrix. 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 | Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions |
| topic | Analysis of PDEs 35K67, 35B45, 47D07, 35J70, 35J75 |
| url | https://arxiv.org/abs/2405.09540 |