A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

被引:11
作者
Briane, M. [1 ]
Casado Diaz, J. [2 ]
机构
[1] INSA Rennes, Inst Rech Math Rennes, Rennes, France
[2] Univ Seville, Dept Ecuac Diferenciales & Anal Numer, Seville, Spain
关键词
Div-curl; Homogenization; Elliptic systems; Non-equi-bounded coefficients; Gamma-convergence; Jacobian; DIFFUSION ENERGIES; FUNCTIONALS; DIRICHLET; EQUATIONS; CLOSURE; SET;
D O I
10.1016/j.jde.2015.12.029
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this paper a new div-curl result is established in an open set Omega of R-N, N >= 2, for the product sigma(n) center dot n(n) of two sequences of vector-valued functions sigma(n,) n(n) such that an is bounded in L-P (Omega)(N), n(n) is bounded in L-q (Omega)(N) with 1/p 1/q = 1 + 1/(N - 1), and such that div sigma(n), curl n(n) are compact in suitable spaces. The new assumption is that the product converges weakly in W--1,W-1 (Omega). The approach is also new in the topic, and is based on a compactness result for bounded sequences in W-1,W-q (Omega) through a suitable selection of annuli on which the gradients are not too high, in the spirit of [26,32] and using the imbedding of W-1,W-q into L-p' for the unit sphere of R-N. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in L-P (Omega) for some p > N-1/2 if N > 2, or in L-1(Omega) if N = 2. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in W-1,W-N-1(Omega) satisfying an alternative assumption to the L-infinity-strong estimate of [8]. Two examples show the sharpness of the results. (C) 2015 Elsevier Inc. All rights reserved.
引用
收藏
页码:5678 / 5725
页数:48
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