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

TL;DR

This paper constructs Green functions and related representations for solutions of elliptic equations, establishing their regularity and continuity properties in the context of weak solutions.

Contribution

It introduces explicit Green functions and integral representations for elliptic PDE solutions with minimal regularity assumptions on the data.

Findings

Green functions are constructed for elliptic equations with measurable coefficients.

Solutions exhibit global Hölder continuity under certain conditions.

Integral representations facilitate analysis of solution regularity.

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 $\int_{\Omega}G(x,y)f(x)dx = u(y)$, $\int_{\Omega}H_{l}(x,y)f(x)dx = -\frac{\ \partial u}{\partial x_{l}}(y)\quad a.e~y\in \Omega, \forall~l\in\{1,\cdots,N\}.$