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

created by csato on 12 Nov 2021

[BibTeX]

preprint

Inserted: 12 nov 2021

Year: 2019

ArXiv: 1901.05783 PDF

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.

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