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

Abstract

In this paper, we characterize all the distributions $F\in {\mathcal{D}}^{\prime }(U)$ such that there exists a continuous weak solution $v\in C(U,{\mathbb{C}}^n)$ (with $U\subset \Omega$) to the divergence-type equation $$L_1^{\ast }{v}_1+...+L_n^{\ast }{v}_n=F,$$ 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.