The equation div$u$+$⟨a,u⟩=f$

Abstract

We study the solutions $u$ to the equation $$\{\begin{array}{ll}\mathrm{div}u+⟨a,u⟩=f& \text{ in }\Omega ,\\ u=0& \text{ on }\partial \Omega ,\end{array}$$ where $a$ and $f$ are given. We significantly improve the existence results of [Csató and Dacorogna, A Dirichlet problem involving the divergence operator, \textit{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 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 Hölder spaces.