On coercivity and irregularity for some nonlinear degenerate elliptic systems

  • Kewei ZhangEmail author
Research Article


We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W 0 1,p which are nowhere C 1 on their supports. We also give examples of universally p-coercive variational integrals in W 0 1,p for p ⩾ with L coefficients for which uniqueminimizers under affine boundary conditions are nowhere C 1.


Degenerate p-Laplace system measurable coefficient universal p-strong coercivity subspace without rank-one matrice nowhere C1 minimizers 


35J45 35J50 35J55 49J10 49N60 


  1. 1.
    Adams R A. Sobolev Spaces. New York: Academic Press, 1975zbMATHGoogle Scholar
  2. 2.
    Astala K. Area distortions of quasiconformal mappings. Acta Math, 1994, 173: 37–60zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Astala K, Faraco D. Quasiregular mappings and Young measures. Proc Royal Soc Edin, 2002, 132A: 1045–1056CrossRefMathSciNetGoogle Scholar
  4. 4.
    Astala K, Iwaniec T, Saksman E. Beltrami operators in the plane. Duke Math J, 2001, 107: 27–56zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ball J M. Convexity conditions and existence theorems in nonlinear elasticity. Arch Rational Mech Anal, 1977, 63: 337–403zbMATHGoogle Scholar
  6. 6.
    Ball J M. A version of the fundamental theorem of Young measures. In: Rascle M, Serre D, Slemrod M, eds. Partial Differential Equations and Continuum Models of Phase Transitions. Berlin: Springer-Verlag, 1989, 207–215Google Scholar
  7. 7.
    Bhattacharya K, Firoozye N B, James R D, Kohn R V. Restrictions on Microstructures. Proc Royal Soc Edin, 1994, 124A: 843–878MathSciNetGoogle Scholar
  8. 8.
    Cellina A, Perrotta S. On a problem of potential wells. J Convex Anal, 1995, 2: 103–115zbMATHMathSciNetGoogle Scholar
  9. 9.
    Ciarlet P G. Mathematical Elasticity. Vol I. Three-dimensional Elasticity. Studies in Mathematics and Its Applications, 20. Amsterdam: North-Holland, 1988CrossRefGoogle Scholar
  10. 10.
    Dacorogna B. Direct Methods in the Calculus of Variations. Berlin: Springer-Verlag, 1989zbMATHGoogle Scholar
  11. 11.
    Dacorogna B, Marcellini P. General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Mathematica, 1997, 178: 1–37zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dacorogna B, Marcellini P. Implicit Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, 37. Boston: Birkhäuser, 1999Google Scholar
  13. 13.
    Dacorogna B, Pisante G. A general existence theorem for differential inclusions in the vector valued case. Port Math (NS), 2005, 62: 421–436zbMATHMathSciNetGoogle Scholar
  14. 14.
    Ekeland I, Temam R. Convex Analysis and Variational Problems. Amsterdam: North-Holland, 1976zbMATHGoogle Scholar
  15. 15.
    Evans L C. Quasiconvexity and partial regularity in the calculus of variations. Arch Rational Mech Anal, 1986, 95: 227–252zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Faraco D, Zhong X. Quasiconvex functions and Hessian equations. Arch Ration Mech Anal, 2003, 168: 245–252zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fonseca I, Müller S, Pedregal P. Analysis of concentration and oscillation effects generated by gradients. SIAM J Math Anal, 1998, 29: 736–756zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Giaquinta M. Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics ETH Zurich. Basel: Birkhäuser Verlag, 1993Google Scholar
  19. 19.
    Gromov M. Partial Differential Relations. Berlin: Springer-Verlag, 1986zbMATHGoogle Scholar
  20. 20.
    Iqbal Z. Variational Methods in Solid Mechanics. Ph D Thesis, University of Oxford, 1999Google Scholar
  21. 21.
    Iwaniec T, Martin G. Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. Oxford: Clarendon Press, 2001Google Scholar
  22. 22.
    Kinderlehrer D, Pedregal P. Characterizations of Young measures generated by gradients. Arch Rational Mech Anal, 1991, 115: 329–365zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kirchheim B. Rigidity and Geometry of Microstructures. MPI for Mathematics in the Sciences Leipzig, Lecture Notes ( note-1603.pdf) 16. 2003
  24. 24.
    Kondratev V A, Oleinik O A. Boundary-value problems for the system of elasticity theory in unbounded domains. Russian Math Survey, 1988, 43: 65–119CrossRefMathSciNetGoogle Scholar
  25. 25.
    Kristensen J. Lower semicontinuity in spaces of weakly differentiable functions. Math Ann, 1999, 313: 653–710zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Leeuw K de, Mirkil H. Majorations dans L des opérateurs différentiels à coefficients constants. C R Acad Sci Paris, 1962, 254: 2286–2288zbMATHMathSciNetGoogle Scholar
  27. 27.
    Morrey C B Jr. Multiple Integrals in The Calculus of Variations. Berlin: Springer, 1966zbMATHGoogle Scholar
  28. 28.
    Müller S. Variational Models for Microstructure and Phase Transitions, 2. Lecture Notes, Max-Planck-Institute for Mathematics in the Sciences, Lepzig. 1998Google Scholar
  29. 29.
    Müller S, Šverák V. Attainment results for the two-well problem by convex integration. In: Jost J, ed. Geometric Analysis and the Calculus of Variations. Boston: International Press, 1996, 239–251Google Scholar
  30. 30.
    Müller S, Šverák V. Unexpected solutions of first and second order partial differential equations. Doc Math J DMV, Extra Vol, ICM 98, 1998, 691–702Google Scholar
  31. 31.
    Müller S, Šverák V. Convex integration with constraints and applications to phase transitions and partial differential equations. J Eur Math Soc, 1999, 1: 393–422zbMATHCrossRefGoogle Scholar
  32. 32.
    Müller S, Šverák V. Convex integration for Lipschitz mappings and counterexamples to regularity. Ann Math, 2003, 157: 715–742zbMATHGoogle Scholar
  33. 33.
    Müller S, Sychev M A. Optimal existence theorems for nonhomogeneous differential inclusions. J Funct Anal, 2001, 181: 447–475zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ornstein D A. Non-inequality for differential operators in the L 1-norm. Arch Rational Mech Anal, 1962, 11: 40–49zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Pedregal P. Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and Their Applications, 30. Basel: Birkhäuser Verlag, 1997Google Scholar
  36. 36.
    Rees E G. Linear spaces of real matrices of large rank. Proc Royal Soc Edin, 1996, 126A: 147–151MathSciNetGoogle Scholar
  37. 37.
    Rockafellar R T. Convex Analysis. Princeton: Princeton University Press, 1970zbMATHGoogle Scholar
  38. 38.
    Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970zbMATHGoogle Scholar
  39. 39.
    Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press, 1993zbMATHGoogle Scholar
  40. 40.
    Šverák V. Rank one convexity does not imply quasiconvexity. Proc Royal Soc Edin, 1992, 120A: 185–189Google Scholar
  41. 41.
    Šverák V. New examples of quasiconvex functions. Arch Rational Mech Anal, 1992, 119: 293–300zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Šverák V. On the problem of two wells. In: Microstructure and Phase Transition, IMA Vol Math Appl, 54. 1994, 183–189Google Scholar
  43. 43.
    Sychev M A. Comparing two methods of resolving homogeneous differential inclusions. Calc Var PDEs, 2001, 13: 213–229zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Tartar L. Compensated compactness and applications to partial differential equations. In: Knops R J, ed. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV. London: Pitman Press, 1979, 136–212Google Scholar
  45. 45.
    Tartar L. Some remarks on separately convex functions. Microstructure and phase transition. IMA Vol Math Appl, 54. New York: Springer, 1993, 191–204Google Scholar
  46. 46.
    Zhang K-W. A construction of quasiconvex functions with linear growth at infinity. Ann Sc Norm Sup Pisa, Serie IV, 1992, XIX: 313–326Google Scholar
  47. 47.
    Zhang K-W. On connected subsets of M 2×2 without rank-one connections. Proc Royal Soc Edin, 1997, 127A: 207–216Google Scholar
  48. 48.
    Zhang K-W. Quasiconvex functions, SO(n) and two elastic wells. Anal Nonlin H Poincaré, 1997, 14: 759–785zbMATHCrossRefGoogle Scholar
  49. 49.
    Zhang K-W. On the structure of quasiconvex hulls. Ann Inst H Poincaré-Analyse Nonlineaire, 1998, 15: 663–686zbMATHCrossRefGoogle Scholar
  50. 50.
    Zhang K-W. Maximal extension for linear spaces of real matrices with large rank. Proc Royal Soc Edin, 2001, 131A: 1481–1491CrossRefGoogle Scholar
  51. 51.
    Zhang K-W. Estimates of quasicovnex polytopes in the calculus of variations. J Convex Anal, 2006, 13: 37–50zbMATHMathSciNetGoogle Scholar
  52. 52.
    Zhang K-W. On coercivity and regularity for linear elliptic systems. Preprint, 2008Google Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsSwansea UniversitySwanseaUK

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