Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Applications of a Weighted Symmetrization Inequality to Elastic Membranes and Plates | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 808693 12 pages doi 2010 808693 Research Article Applications of a Weighted Symmetrization Inequality to Elastic Membranes and Plates Behrouz Emamizadeh Department of Mathematics The Petroleum Institute . Box 2533 Abu Dhabi United Arab Emirates Correspondence should be addressed to Behrouz Emamizadeh bemamizadeh@ Received 28 January 2010 Accepted 10 June 2010 Academic Editor Marta Garcia-Huidobro Copyright 2010 Behrouz Emamizadeh. 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. This paper is devoted to some applications of a weighted symmetrization inequality related to a second order boundary value problem. We first interpret the inequality in the context of elastic membranes and observe that it lends itself to make a comparison between the deflection of a membrane with a varying density with that of a membrane with a uniform density. Some mathematical consequences of the inequality including a stability result are presented. Moreover a similar inequality where the underlying differential equation is of fourth order is also discussed. 1. Introduction In this paper we discuss some applications of a weighted symmetrization inequality related to a second-order boundary value problem. We begin by interpreting the inequality in the context of elastic membranes. Let us briefly describe the physical situation and its mathematical formulation that leads to the inequality we are interested in. An elastic membrane of varying density a x is occupying a region Q a disk in the plane R2. The membrane is fixed at the boundary and is subject to a load f x h x . The governing equation in terms of the deflection function u x is the elliptic boundary value problem -V a x Vu f x h x in Q J P u 0 on dQ On the .
