An elementary proof of the continuity from L20(Ω) to H10(Ω)n of Bogovskii’s right inverse of the divergence

Abstract

The existence of right inverses of the divergence as an operator from H1 0 (Ω)n to L 2 0(Ω) is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When Ω is a bounded domain which is star-shaped with respect to a ball B, a right inverse given by an integral operator was introduced by Bogovskii, who also proved its continuity using the Calder´on-Zygmund theory of singular integrals. In this paper we give an alternative elementary proof of the continuity using the Fourier transform. As a consequence, we obtain estimates for the constant in the continuity in terms of the ratio between the diameter of Ω and that of B. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincar´e inequalities, we obtain estimates for the constants in these two inequalities. We also show that one can proceed in the opposite way, that is, the existence of a continuous right inverse of the divergence, as well as estimates for the constant in that continuity, can be obtained from the improved Poincar´e inequality. We give an interesting example of this situation in the case of convex domains.