Green representations and global Hölder continuity for solutions of elliptic equations

Let $N\in \mathbb{N}$ and $u$ be a weak solution of equation ${\displaystyle Lu\equiv -\sum _{i,j=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}(\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}{b}^{ij})=f}$ in $\mathrm{\Omega }\subset {\mathbb{R}}^N$. We obtain functions $G$ and $H_l$ on $\mathrm{\Omega }\times \mathrm{\Omega }$ for every $l\in \{1,\cdots ,N\}$ having following properties: if $f$ is in $L^1(\mathrm{\Omega })$, then ${\int }_{\mathrm{\Omega }}G(x,y)f(x)dx=u(y)$, ${\int }_{\mathrm{\Omega }}{H}_l(x,y)f(x)dx=-\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_l}(y)a\mathrm{.}ey\in \mathrm{\Omega },\mathrm{\forall }l\in \{1,\cdots ,N\}\mathrm{.}$