An elementary proof of the continuity from L ²₀(Ω) to H ¹₀(Ω) ⁿ of Bogovskii’s right inverse of the divergence

摘要

The existence of right inverses of the divergence as an operator from H 1 0 (.) n to L 2 0 (.) is a problem that has been widely studied because of its importance in the analysis of the classic equations of .uid dynamics. When . 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 . 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