JournalsaihpcVol. 17, No. 2pp. 193–217

Global higher integrability of Jacobians on bounded domains

  • Chun Li

    School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia
  • Alan McIntosh

    School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia
  • Kewei Zhang

    School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia
  • Jeff Hogan

    School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia
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Abstract

We give conditions for a vector-valued function u ∈ W1,n(Ω, ℝn), satisfying det Du(x) ≥ 0 on a bounded domain Ω, which imply that det Du(x) is globally higher integrable on Ω. We also give conditions for u ∈ W1,n (Ω, ℝn) such that det Du belongs to the Hardy space h1z(Ω) and exhibit some examples which show that our conditions are in some sense optimal. Applications to the weak convergence of Jacobians follow. Div-curl type extensions of these results to forms are also considered.

Résumé

Pour une fonction à valeurs vectorielles u ∈ W1,n(Ω, ℝn) telle que det Du(x) ≥ 0 dans un ouvert borné Ω, nous donnons des conditions conduisant à une amélioration de l'intégrabilité globale de det Du(x) dans un ouvert borné Ω. Nous donnons aussi des conditions sur u ∈ W1,n(Ω, ℝn) pour que det Du appartienne à l'espace de Hardy h1z(Ω). Quelques exemples démontrent que ces conditions sont dans un certain sens optimales. Ces résultats sont appliqués à la convergence faible des jacobiens. Nous examinons aussi l'extension de ces résultats du type div-curl aux formes différentielles.

Cite this article

Chun Li, Alan McIntosh, Kewei Zhang, Jeff Hogan, Global higher integrability of Jacobians on bounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 2, pp. 193–217

DOI 10.1016/S0294-1449(00)00108-6