# Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

## TL;DR

This paper characterizes distributions that admit continuous weak solutions to divergence-type equations associated with elliptic systems of complex vector fields, extending classical results to more general elliptic systems.

## Contribution

It provides a comprehensive characterization of distributions for which divergence-type equations have continuous solutions in the context of complex elliptic systems.

## Key findings

- Characterization of distributions with continuous solutions
- Extension of classical divergence results to complex elliptic systems
- Unified framework for divergence equations with elliptic vector fields

## Abstract

In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset \Omega$) to the divergence-type equation $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F,$$ where $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $\Omega \subset \mathbb{R}^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\mathbb{R}^N$, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer.