Abstract

We prove partial regularity of minimizers for a class of polyconvex integral functionals ∫Ωf(Du,Ad Du,det Du) dx,\int_\Omega f (Du, \text{Ad}\, Du, \text{det}\, Du)\, dx, where ff is degenerate convex. Our class includes the model case ∫Ω(∣Du∣p+∣Ad Du∣p+∣det Du∣p) dx.\int_\Omega (|Du|^p + |\text{Ad}\, Du|^p + |\text{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 ∫Ω∣Du∣p dx\int_\Omega |Du|^p\, dx.

Author information

Contact details are reproduced from the original publication and may be historical.

Luca Esposito

Dip. di Ingegneria dell'Informazione e Matematica Applicata, Università di Salerno, Italy

Giuseppe Mingione

Dip. di Matematica, Università di Parma, Via D'Azeglio 85/a, 43100 Parma, Italy

Keywords

  • Polyconvexity
  • regularity
  • elliptic systems

Mathematics Subject Classification

49N60; 49N99, 35J20

References

  1. E. Acerbi, G. Dal Maso: New lower semicontinuity results for polyconvex integrals, Calc. Var. 2 (1994) 329–371.https://doi.org/10.1007/BF01235534Google Scholar
  2. E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1984) 125–145.https://doi.org/10.1007/BF00275731Google Scholar
  3. E. Acerbi, N. Fusco: A regularity theorem for minimizers of quasiconvex integrals, Arch. Rat. Mech. Anal. 99 (1987) 261–281.https://doi.org/10.1007/BF00284509Google Scholar
  4. E. Acerbi, N. Fusco: Partial regularity under anisotropic (p, q) growth conditions, J. Diff. Equat. 107 (1994) 46–47.https://doi.org/10.1006/jdeq.1994.1002Google Scholar
  5. E. Acerbi, G. Mingione: Regularity results for a class of quasiconvex functionals with nonstandard growth, to appear.Google Scholar
  6. E. Acerbi, G. Mingione: Functionals with p(x) growth and regularity, Atti Accad. Naz. Lincei Cl. Sci. Mat. Natur. Rend. Lincei (9) Mat. Appl., to appear.Google Scholar
  7. J. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 69 (1977) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  8. J. Ball, Murat F.: W 1,p -quasiconvexity and variational problems for multiple integrals, J. Functional Analysis 58 (1984) 225–253.https://doi.org/10.1016/0022-1236(84)90041-7Google Scholar
  9. M. Carozza, N. Fusco, G. Mingione: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura e Appl. 175(IV) (1998) 141–164.https://doi.org/10.1007/BF01783679Google Scholar
  10. B. Dacorogna: Direct Methods in the Calculus of Variations, Appl. Math. Sci. 78, Springer-Verlag, 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  11. B. Dacorogna, P. Marcellini: Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris 311 (1990) 393–396.Google Scholar
  12. L. Evans: Quasiconvexity and partial regularity in the calculus of variations, Arch. Rat. Mech. Anal. 95 (1984) 227–252.https://doi.org/10.1007/BF00251360Google Scholar
  13. L. Evans, R. Gariepy: Blow-up, compactness and partial regularity in the calculus of variations, Indiana U. Math. J. 36 (1987) 361–371.https://doi.org/10.1512/iumj.1987.36.36021Google Scholar
  14. I. Fonseca, N. Fusco: Regularity results for anisotropic image segmentation models, Ann. Scu. Norm. Sup. Pisa Cl. Sci. 24(IV) (1997) 463–499.Google Scholar
  15. I. Fonseca, J. Malý: Relaxation of multiple integrals below the growth exponent for the energy density, Ann. IHP (Analyse non lineare) 14(3) (1997) 309–338.https://doi.org/10.1016/s0294-1449(97)80139-4Google Scholar
  16. M. Fuchs: Regularity for a class of variational integrals motivated by nonlinear elasticity, Asymptotic Anal. 9 (1994) 181–197.https://doi.org/10.3233/ASY-1994-9102Google Scholar
  17. M. Fuchs, G. Seregin: Hölder continuity for weak extremals of some two-dimensional variational problems related to nonlinear elasticity, Adv. Math. Sci. Appl. 7 (1997) 413–125.Google Scholar
  18. M. Fuchs, G. Seregin: Partial regularity of the deformation gradient for some model problems in nonlinear two-dimensional elasticity, Algebra i Analiz (St. Petersburg Math. J.) 6 (1994) 128–153.Google Scholar
  19. N. Fusco, J. Hutchinson: C 1,α partial regularity of functions minimizing quasiconvex integrals, Manuscripta Math. 54 (1985) 121–143.https://doi.org/10.1007/BF01171703Google Scholar
  20. N. Fusco, J. Hutchinson: Partial regularity for minimizers of certain functionals having nonquadratic growth, Ann. Mat. Pura e Appl. 155(IV) (1989) 1–24.https://doi.org/10.1007/BF01765932Google Scholar
  21. N. Fusco, J. Hutchinson: Partial regularity in problems motivated by nonlinear elasticity, SIAM J. Math. Anal. 22 (1991) 1516–1551.https://doi.org/10.1137/0522098Google Scholar
  22. N. Fusco, J. Hutchinson: Partial regularity and everywhere continuity for a model problem from nonlinear elasticity, J. Austr. Math. Soc. 57(2) (1994) 158–169.https://doi.org/10.1017/S1446788700037496Google Scholar
  23. N. Fusco, J. Hutchinson: A direct proof of lower semicontinuity of polyconvex integrals, Manuscripta Math. 87(2) (1995) 561–565.https://doi.org/10.1007/BF02570460Google Scholar
  24. M. Giaquinta: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies 105, Princeton Univ. Press, 1983.https://doi.org/10.1515/9781400881628Google Scholar
  25. M. Giaquinta, G. Modica: Partial regularity of minimizers of quasi-convex integrals, Ann. IHP (Analyse non lineare) 3 (1986) 185–208.https://doi.org/10.1016/s0294-1449(16)30385-7Google Scholar
  26. M. Giaquinta, G. Modica: Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math. 57 (1986) 55–90.https://doi.org/10.1007/BF01172492Google Scholar
  27. M. Giaquinta, G. Modica, J. Suocek: Cartesian currents, weak diffeomorphisms and existence theorems in the nonlinear elasticity, Arch. Rat. Mech. Anal. 106 (1989) 97–159.https://doi.org/10.1007/BF00251429Google Scholar
  28. E. Giusti: Metodi Diretti nel Calcolo delle Variazioni, U.M.I., Bologna, 1994.Google Scholar
  29. C. Hamburger: Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelle’s J.) 431 (1992) 7–64.https://doi.org/10.1515/crll.1992.431.7Google Scholar
  30. J. Malý: Weak lower semicontinuity of polyconvex integrals, Proc. Royal Soc. Edinb. 123 A (1993) 681–691.https://doi.org/10.1017/S0308210500030900Google Scholar
  31. P. Marcellini: On the definition and lower semicontinuity of certain quasiconvex integrals, Ann. IHP (Analyse non lineare) 3 (1986) 391–409.https://doi.org/10.1016/s0294-1449(16)30379-1Google Scholar
  32. P. Marcellini: Everywhere regularity for a class of elliptic systems withouth growth conditions, Ann. Scu. Norm. Sup. Pisa Cl. Sci. 23(IV) (1996) 1–25.Google Scholar
  33. P. Marcellini: Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), 1–30.https://doi.org/10.1016/0022-0396(91)90158-6Google Scholar
  34. P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions Arch. Rat. Mech. Anal. 105 (1989) 267–284.https://doi.org/10.1007/BF00251503Google Scholar
  35. S. Muller: Weak continuity of determinant and nonlinear elasticity, C. R. Acad. Sci. Paris 307 (1988) 501–506.Google Scholar
  36. S. Muller, J. Sivaloganathan, S. J. Spector: An isoperimetric estimate and W 1,p - quasiconvexity in nonlinear elasticity, Calc. Var. 8 (1999) 159–176.https://doi.org/10.1007/s005260050121Google Scholar
  37. K. Uhlenbeck: Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977) 219–240.https://doi.org/10.1007/BF02392316Google Scholar

Suggested citation

L. Esposito, G. Mingione. “Partial Regularity for Minimizers of Degenerate Polyconvex Energies.” Journal of Convex Analysis 8 (2001), No. 1, 1–38. https://doi.org/10.68381/jca08001

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2001. Rights now held by Banach Press LLC.