Green representations and global H\"older continuity for solutions of elliptic equations

Abstract

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 $\Omega\subset \mathbb{R}^{N}$. We obtain functions $G$ and $H_{l}$ on $\Omega\times \Omega$ for every $l\in\{1,\cdots,N\}$ having following properties: if $f$ is in $L^{1}(\Omega)$, then $\int_{\Omega}G(x,y)f(x)dx = u(y)$, $\int_{\Omega}H_{l}(x,y)f(x)dx = -\frac{\ \partial u}{\partial x_{l}}(y)\quad a.e~y\in \Omega, \forall~l\in\{1,\cdots,N\}.$