Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909328803364864 |
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| author | Dong, Hongjie Yang, Dachun Yang, Sibei |
| author_facet | Dong, Hongjie Yang, Dachun Yang, Sibei |
| contents | Let $n \ge 2$ and $Ω\subset \mathbb{R}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $Ω$ for weak solutions to the second-order elliptic equation $\mathrm{div}(A \nabla u) = \mathrm{div} , \boldsymbol{f}$ in $Ω$. This is achieved under minimal regularity assumptions on $Ω$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $Ω$. As an application, we also obtain global estimates of $\nabla u$ in the Lebesgue space $L^1(Ω)$ when $\boldsymbol{f}$ belongs to the Hardy space on $Ω$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19498 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains Dong, Hongjie Yang, Dachun Yang, Sibei Analysis of PDEs Functional Analysis 35J25 35B45 Let $n \ge 2$ and $Ω\subset \mathbb{R}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $Ω$ for weak solutions to the second-order elliptic equation $\mathrm{div}(A \nabla u) = \mathrm{div} , \boldsymbol{f}$ in $Ω$. This is achieved under minimal regularity assumptions on $Ω$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $Ω$. As an application, we also obtain global estimates of $\nabla u$ in the Lebesgue space $L^1(Ω)$ when $\boldsymbol{f}$ belongs to the Hardy space on $Ω$. |
| title | Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains |
| topic | Analysis of PDEs Functional Analysis 35J25 35B45 |
| url | https://arxiv.org/abs/2409.19498 |