Green representations and global Hölder continuity for solutions of elliptic equations
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918349595738112 |
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| author | Duong, Duc |
| author_facet | Duong, Duc |
| contents | Let $N\in\mathbb{N}$ and $u$ be a weak solution of equation $\displaystyle Lu\equiv - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{j}}(\frac{\partial u}{\partial x_{i}}b^{ij})= f$ in $Ω\subset \mathbb{R}^{N}$. We obtain functions $G$ and $H_{l}$ on $Ω\times Ω$ for every $l\in\{1,\cdots,N\}$ having following properties: if $f$ is in $L^{1}(Ω)$, then $\int_ΩG(x,y)f(x)dx = u(y)$, $\int_ΩH_{l}(x,y)f(x)dx = -\frac{\ \partial u}{\partial x_{l}}(y)\quad a.e~y\in Ω, \forall~l\in\{1,\cdots,N\}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Green representations and global Hölder continuity for solutions of elliptic equations Duong, Duc Analysis of PDEs 35J15, 35J08, 35B65, 35J70, 35J75 Let $N\in\mathbb{N}$ and $u$ be a weak solution of equation $\displaystyle Lu\equiv - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{j}}(\frac{\partial u}{\partial x_{i}}b^{ij})= f$ in $Ω\subset \mathbb{R}^{N}$. We obtain functions $G$ and $H_{l}$ on $Ω\times Ω$ for every $l\in\{1,\cdots,N\}$ having following properties: if $f$ is in $L^{1}(Ω)$, then $\int_ΩG(x,y)f(x)dx = u(y)$, $\int_ΩH_{l}(x,y)f(x)dx = -\frac{\ \partial u}{\partial x_{l}}(y)\quad a.e~y\in Ω, \forall~l\in\{1,\cdots,N\}.$ |
| title | Green representations and global Hölder continuity for solutions of elliptic equations |
| topic | Analysis of PDEs 35J15, 35J08, 35B65, 35J70, 35J75 |
| url | https://arxiv.org/abs/2602.18737 |