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Partial Regularity for Minimizers of Degenerate Polyconvex Energies

Partial Regularity for Minimizers of Degenerate Polyconvex Energies

Partial Regularity for Minimizers of Degenerate Polyconvex

Journal of Convex Analysis Volume 8 (2001), No. 1, 1–38 Partial Regularity for Minimizers of Degenerate Polyconvex Energies ∗ Luca Esposito Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università di Salerno, Italy. e-mail: posito@matna2.dma.unina.it Giuseppe Mingione Dipartimento di Matematica dell’ Università di Parma, Via D’Azeglio 85/a, 43100 Parma, Italy. e-mail: mingione@prmat.math.unipr.it Received October 27, 1999 Revised manuscript received July 24, 2000 We prove partial regularity of minimizers for a class of polyconvex integral functionals ∫ Ω f(Du, Ad Du, det Du) dx, where f is degenerate convex. Our class includes the model case ∫ Ω (| Du | p + | Ad Du | p + | det Du | p ) dx. The method of proof involves a blow-up technique combined with a suitable asymptotic analysis of the degeneration nature of the first term ∫ | Du |p dx. Ω Keywords: Polyconvexity, Regularity, Elliptic Systems 1991 Mathematics Subject Classification: 49N60, 49N99, 35J20 Contents 1. Introduction 2 2. Some multilinear algebra 4 3. The class ∧ k W 1,p (Ω) and the existence of minimizers 6 4. Preliminary results and notation 9 5. Statement of the alternative 10 6. Decay estimate in the first case 14 7. Decay estimate in the second case 20 8. Preliminary estimates for strong convergences 22 9. Weak convergences turn into strong convergences 27 10. Proof of the Main Theorem 34 11. Final Remarks 35 References 36 ∗ This work has been performed as a part of the Research Project “Modelli Variazionali sotto Ipotesi Non standard” supported by GNAFA-CNR ISSN 0944-6532 / $ 2.50 c○ Heldermann Verlag

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