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arXiv · 1901.05783

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

Pierre Bousquet·Gyula Csató
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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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Pierre Bousquet, Gyula Csató. 2019-01-17. The equation div$u$+$\langle a, u \rangle=f$. https://doi.org/10.1142/s0219199719500354

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