Real analytic solutions to the divergence equation

Abstract

In this paper, we develop a differential-topological method to yield explicit real analytic solutions $v$ to the divergence equation $di{v}_{{\mathbb{R}}^n}v=f$ on any annali $A(R_1,R_2)=\{x\in {\mathbb{R}}^n:R_1<|x|<R_2\}$, with $n\ge 2$, and $0<R_1<R_2<\infty$. The prescribed source term $f$ is supposed to be real analytic on $\overline{A(R_1,R_2)}=\{x\in {\mathbb{R}}^n:R_1\le |x|\le R_2\}$ satisfying the zero integral condition on $A(R_1,R_2)$. The resulting solution $v$ is a real analytic vector field on $\overline{A(R_1,R_2)}$, which vanishes on $\partial (A(R_1,R_2))$. The method which we develop here is different from the standard Bogovski approach and the Kapitanskii-Pileckas approach. The first main step our method is a clever differential-topological argument, which we develop under the inspiration and guidance of the standard proof of the cohomological statement $H_c^n\left({\mathbb{R}}^n\right)=\mathbb{R}$ in Spviak book A Comprehensive Introduction to Differential Geometry, Vol I. This allows us to reduce the problem to that of solving a linear algebra problem.

26 pages