Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908392085258240 |
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| author | Dong, Hongjie Ryu, Junhee |
| author_facet | Dong, Hongjie Ryu, Junhee |
| contents | We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. Additionally, they have small bounded mean oscillations in the other spatial variables. We obtain the well-posedness and regularity of solutions in weighted mixed-norm Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients Dong, Hongjie Ryu, Junhee Analysis of PDEs 35J70, 35K65, 35D30, 35R05 We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. Additionally, they have small bounded mean oscillations in the other spatial variables. We obtain the well-posedness and regularity of solutions in weighted mixed-norm Sobolev spaces. |
| title | Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients |
| topic | Analysis of PDEs 35J70, 35K65, 35D30, 35R05 |
| url | https://arxiv.org/abs/2412.00779 |