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: Weak lower semicontinuity of variational functionals with variable growth | Yongqiang Journal of Inequalities and Applications 2011 2011 19 http content 2011 1 19 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Weak lower semicontinuity of variational functionals with variable growth Fu Yongqiang Correspondence fuyqhagd@yahoo. cn Department of Mathematics Harbin Institute of Technology Harbin 150001 China SpringerOpen0 Abstract In this paper we establish the weak lower semicontinuity of variational functionals with variable growth in variable exponent Sobolev spaces. The weak lower semicontinuity is interesting by itself and can be applied to obtain the existence of an equilibrium solution in nonlinear elasticity. 2000 Mathematics Subject Classification 49A45 Keywords lower semicontinuity variational functional variable growth 1 Introduction The main purpose of this paper is to study the weak lower semicontinuity of the functional f0. . V w where i is a bounded C 1 domain in Rn and f Rn X Rm X Rnm R is a Caratheod-ory function satisfying variable growth conditions. If m n 1 Tonelli 1 proved that the functional F is lower semicontinuity in W1 a b if and only if the function f is convex in the last variable. Later several authors generalized this result in which x is allowed to belong to Rn with n 1 see for example Serrin 2 and Marcellini and Sbordone 3 . On the other hand if we allow the function u to be vector-valued . m 1 then the convexity hypothesis turns to be sufficient but unnecessary. A suitable condition termed quasiconvex was introduced by Morrey 4 . Morrey showed that under strong regularity assumptions on f F is weakly lower semicontinuous in W1 O Rm if and only iff is quasiconvex in the last variable. Afterward for f satisfying so-called natural growth condition 0 f x z. Ệ a x C izlp ỊỆ p where p 1 C 0 and a x 0 are locally integrable Acerbi and Fusco 5 proved that F is weakly lower semicontinuous in W1 p i Rm if and only if f is quasiconvex in
