2011/03/18 by Ricardo G. Durán, Duran, Ricardo G. · 3 citations
Physics and Astronomy · Mathematics · Computer Science · #Advanced Mathematical Theories and Applications #Mathematical and Theoretical Analysis #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1103.3718
The existence of right inverses of the divergence as an operator form H10(Ω)n to L02(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 the continuity using the Calderón-Zygmund theory of singular integrals. In this paper we give an alternative elementary proof using the Fourier transform. As a consequence, we obtain estimates of the constant in the continuity in terms of the ratio between the diameters of Ω and B. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincaré inequalities, we obtain estimates for the constants in these two inequalities.