2015/01/09 by Marc Briane, Briane, M., Juan Casado‐Díaz +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.1501.02152
openalex publication_date 2015/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper a new div-curl result is established in an open set Ω of ℝN, N≥ 2, for the product of two sequences of vector-valued functions which are bounded respectively in Lp(Ω)N and Lq(Ω)N, with 1/p+1/q=1+1/(N-1), and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product converges weakly in W-1,1(Ω). The key ingredient of the proof is a compactness result for bounded sequences in W1,q(Ω), based on the imbedding of W1,q(S_N-1) into Lp'(S_N-1) (S_N-1 the unit sphere of ℝN) through a suitable selection of annuli on which the gradients are not too high, in the spirit of De Giorgi and Manfredi. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in Lρ(Ω) for some ρ\textgreaterN-1\over 2 if N\textgreater2, or in L1(Ω) if N=2. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in W1,N-1(Ω) satisfying an alternative assumption to the L^∞-strong estimate of Brezis and Nguyen. Two examples show the sharpness of the results.