Green representations and global Hölder continuity for solutions of elliptic equations

Abstract

Let $N\in \mathbb{N}$ and $u$ be a weak solution of equation $Lu\equiv -\sum _{i,j=1}^N\frac{\partial }{\partial x_j}(\frac{\partial u}{\partial x_i}{b}^{ij})=f$ 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 ${\int }_{\Omega }G(x,y)f(x)dx=u(y)$, ${\int }_{\Omega }{H}_l(x,y)f(x)dx=-\frac{\partial u}{\partial x_l}(y)a.ey\in \Omega ,\forall l\in \{1,\cdots ,N\}.$

arXiv admin note: text overlap with arXiv:2509.23064 by other authors