# The equation div$u$+$\langle a, u \rangle=f$

## TL;DR

This paper improves existence results for solutions to a divergence equation with a linear term, establishing sharp conditions on the coefficients and covering Sobolev and Hölder spaces.

## Contribution

It provides the first comprehensive existence results under optimal regularity assumptions for the divergence equation involving a vector field and a source term.

## Key findings

- Existence of solutions under sharp regularity conditions on coefficients.
- Necessary and sufficient condition on the vector field a.
- Results applicable to Sobolev and Hölder spaces.

## Abstract

We study the solutions $u$ to the equation $$ \begin{cases} \operatorname{div} u + \langle a , u \rangle = f & \textrm{ in } \Omega,\\ u=0 & \textrm{ on } \partial \Omega, \end{cases} $$ where $a$ and $f$ are given. We significantly improve the existence results of [Csat\'o and Dacorogna, A Dirichlet problem involving the divergence operator, \textit{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 33 (2016), 829--848], where this equation has been considered for the first time. In particular, we prove the existence of a solution under essentially sharp regularity assumptions on the coefficients. The condition that we require on the vector field $a$ is necessary and sufficient. Finally, our results cover the whole scales of Sobolev and H\"older spaces.