TAILIEUCHUNG - Báo cáo sinh học: " Research Article Global Optimal Regularity for the Parabolic Polyharmonic Equations"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Global Optimal Regularity for the Parabolic Polyharmonic Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 879821 12 pages doi 2010 879821 Research Article Global Optimal Regularity for the Parabolic Polyharmonic Equations Fengping Yao Department of Mathematics Shanghai University Shanghai 200436 China Correspondence should be addressed to Fengping Yao yfp1123@ Received 21 February 2010 Accepted 3 June 2010 Academic Editor Vicentiu D. Radulescu Copyright 2010 Fengping Yao. 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. We show the global regularity estimates for the following parabolic polyharmonic equation ut -A mu f in R X 0 TO m e Z under proper conditions. Moreover it will be verified that these conditions are necessary for the simplest heat equation ut - ầu f in R X 0 to . 1. Introduction Regularity theory in PDE plays an important role in the development of second-order elliptic and parabolic equations. Classical regularity estimates for elliptic and parabolic equations consist of Schauder estimates Lp estimates De Giorgi-Nash estimates Krylov-Safonov estimates and so on. Lp and Schauder estimates which play important roles in the theory of partial differential equations are two fundamental estimates for elliptic and parabolic equations and the basis for the existence uniqueness and regularity of solutions. The objective of this paper is to investigate the generalization of Lp estimates that is regularity estimates in Orlicz spaces for the following parabolic polyharmonic problems ut x f -A mu x f f x f in R X 0 to u x 0 0 in R where x x1 . xn A 2 1 d2 dx2 and m is a positive integer. Since the 1960s the need to use wider spaces of functions than Sobolev spaces arose out of various practical problems. Orlicz spaces have been studied as the generalization of Sobolev spaces since they were .

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