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Báo cáo hóa học: "Research Article ´ Brezis-Wainger Inequality on Riemannian Manifolds"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article ´ Brezis-Wainger Inequality on Riemannian Manifolds | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 715961 6 pages doi 10.1155 2008 715961 Research Article Brezis-Wainger Inequality on Riemannian Manifolds PrzemysJaw Gorka Department of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00661 Warsaw Poland Correspondence should be addressed to Przemysiaw Gorka pgorka@mini.pw.edu.pl Received 10 November 2007 Accepted 29 April 2008 Recommended by Shusen Ding The Brezis-Wainger inequality on a compact Riemannian manifold without boundary is shown. For this purpose the Moser-Trudinger inequality and the Sobolev embedding theorem are applied. Copyright 2008 Przemyslaw Gorka. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction There is no doubt that the Brezis-Wainger inequality see 1 is a very useful tool in the examination of partial differential equations. Namely a lot of estimates to a solution of PDE are obtained with the help of the Brezis-Wainger inequality. Especially the inequality is often applied in the theory of wave maps. In this paper we extend the Brezis-Wainger result onto a compact Riemannian manifold. We show the following theorem. Theorem 1.1. Let M g be an n-dimensional compact Riemannian manifold and u G Hk p M M_udVg 0 for n k n p where k is a positive integer and p 1 is a real number. Then u G L M and Mhv M C log ZML llullHk n k M where C C k Mf is a positive constant. 1.1 The proof relies on the application of a Moser-Trudinger inequality see Theorem 2.2 and the Sobolev embedding theorem see Theorem 2.1 . Moreover we will use the integral representation of a smooth function via the Green function see 2 . 2 Journal of Inequalities and Applications 2. Preliminaries In order to make this paper more readable we recall some definitions and facts .