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 $div_{\mathbb{R}^n} v = f$ on any annali $A(R_1 ,R_2) = \{ x \in \mathbb{R}^n : R_1 < |x| < R_2\}$, with $n \geq 2$, and $0 < R_1 < R_2 < \infty$. The prescribed source term $f$ is supposed to be real analytic on $\overline{A(R_1 , R_2)} = \{ x \in \mathbb{R}^n : R_1 \leq |x| \leq R_2\}$ 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 $\partial \big( A(R_1, R_2 ) \big )$. 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 $H_c^n \big ( \mathbb{R}^n\big ) = \mathbb{R}$ 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.