On coercivity and irregularity for some nonlinear degenerate elliptic systems

Abstract

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 W₀^1,p which are nowhere C¹ on their supports. We also give examples of universally p-coercive variational integrals in W₀^1,p for p ≥ 2 with L^∞ coefficients for which unique minimizers under affine boundary conditions are nowhere C¹.

References

1. Adams R A . Sobolev Spaces. New York: Academic Press, 1975 2. Astala K . Areadistortions of quasiconformal mappings. Acta Math, 1994, 173: 37–60. doi:10.1007/BF02392568 3. Astala K, Faraco D . Quasiregular mappings andYoung measures. Proc Royal Soc Edin, 2002, 132A: 1045–1056. doi:10.1017/S0308210500002006 4. Astala K, Iwaniec T, Saksman E . Beltrami operators in the plane. Duke Math J, 2001, 107: 27–56. doi:10.1215/S0012-7094-01-10713-8 5. Ball J M . Convexity conditions and existence theorems in nonlinear elasticity. Arch Rational Mech Anal, 1977, 63: 337–403 6. Ball J M . A version of the fundamental theorem of Young measures. In: Rascle M, SerreD, Slemrod M, eds. Partial Differential Equations and Continuum Models of Phase Transitions. Berlin: Springer-Verlag, 1989, 207–215 7. Bhattacharya K, Firoozye N B, James R D, Kohn R V . Restrictionson Microstructures. Proc Royal Soc Edin, 1994, 124A: 843–878 8. Cellina A, Perrotta S . On a problem of potentialwells. J Convex Anal, 1995, 2: 103–115 9. Ciarlet P G . Mathematical Elasticity. Vol I. Three-dimensional Elasticity. Studies in Mathematicsand Its Applications, 20. Amsterdam: North-Holland, 1988 10. Dacorogna B . DirectMethodsin the Calculus of Variations. Berlin: Springer-Verlag, 1989 11. Dacorogna B, Marcellini P . General existence theoremsfor Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Mathematica, 1997, 178: 1–37. doi:10.1007/BF02392708 12. Dacorogna B, Marcellini P . Implicit Partial DifferentialEquations. Progress in Nonlinear DifferentialEquations and Their Applications, 37. Boston: Birkhäuser, 1999 13. Dacorogna B, Pisante G . A general existence theoremfor differential inclusions in the vector valued case. Port Math (NS), 2005, 62: 421–436 14. Ekeland I, Temam R . Convex Analysis and VariationalProblems. Amsterdam: North-Holland, 1976 15. Evans L C . Quasiconvexity and partial regularity in the calculus of variations. Arch Rational Mech Anal, 1986, 95: 227–252. doi:10.1007/BF00251360 16. Faraco D, Zhong X . Quasiconvex functions andHessian equations. Arch Ration Mech Anal, 2003, 168: 245–252 17. Fonseca I, Müller S, Pedregal P . Analysis of concentration and oscillation effects generatedby gradients. SIAM J Math Anal, 1998, 29: 736–756. doi:10.1137/S0036141096306534 18. Giaquinta M . Introductionto Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics ETH Zurich. Basel: Birkhäuser Verlag, 1993 19. Gromov M . PartialDifferential Relations. Berlin: Springer-Verlag, 1986 20. Iqbal Z . VariationalMethods in Solid Mechanics. Ph D Thesis,University of Oxford, 1999 21. Iwaniec T, Martin G . Geometric Function Theoryand Non-linear Analysis. Oxford MathematicalMonographs. Oxford: Clarendon Press, 2001 22. Kinderlehrer D, Pedregal P . Characterizations of Youngmeasures generated by gradients. Arch RationalMech Anal, 1991, 115: 329–365. doi:10.1007/BF00375279 23. Kirchheim B . Rigidityand Geometry of Microstructures. MPI forMathematics in the Sciences Leipzig, Lecture Notes 16. 2003 24. Kondratev V A, Oleinik O A . Boundary-value problems forthe system of elasticity theory in unbounded domains. Russian Math Survey, 1988, 43: 65–119. doi:10.1070/RM1988v043n05ABEH001945 25. Kristensen J . Lowersemicontinuity in spaces of weakly differentiable functions. Math Ann, 1999, 313: 653–710. doi:10.1007/s002080050277 26. Leeuw K de, Mirkil H . Majorations dans L_∞ des opérateursdifférentiels è coefficients constants. C R Acad Sci Paris, 1962, 254: 2286–2288 27. Morrey C B Jr . Multiple Integrals in The Calculus of Variations. Berlin: Springer, 1966 28. Müller S . VariationalModels for Microstructure and Phase Transitions, 2. Lecture Notes, Max-Planck-Institute for Mathematics in the Sciences,Lepzig. 1998 29. Müller S, Šverák V . Attainmentresults for the two-well problem by convex integration.In: Jost J, ed. GeometricAnalysis and the Calculus of Variations. Boston: International Press, 1996, 239–251 30. Müller S, Šverák V . Unexpectedsolutions of first and second order partial differential equations. Doc Math J DMV, Extra Vol, ICM 98, 1998, 691–702 31. Müller S, Šverák V . Convex integrationwith constraints and applications to phase transitions and partialdifferential equations. J Eur Math Soc, 1999, 1: 393–422. doi:10.1007/s100970050012 32. Müller S, Šverák V . Convex integrationfor Lipschitz mappings and counterexamples to regularity. Ann Math, 2003, 157: 715–742 33. Müller S, Sychev M A . Optimal existence theoremsfor nonhomogeneous differential inclusions. J Funct Anal, 2001, 181: 447–475. doi:10.1006/jfan.2000.3726 34. Ornstein D A . Non-inequality for differential operators in the L¹-norm. Arch Rational Mech Anal, 1962, 11: 40–49. doi:10.1007/BF00253928 35. Pedregal P . Parametrizedmeasures and variational principles. Progressin Nonlinear Differential Equations and Their Applications, 30. Basel: BirkhäuserVerlag, 1997 36. Rees E G . Linear spaces of real matrices of large rank. Proc Royal Soc Edin, 1996, 126A: 147–151 37. Rockafellar R T . Convex Analysis. Princeton: Princeton University Press, 1970 38. Stein E M . Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970 39. Stein E M . Harmonic Analysis: Real-Variable Methods, Orthogonality, and OscillatoryIntegrals. Princeton: Princeton University Press, 1993 40. Šverák V . Rankone convexity does not imply quasiconvexity. Proc Royal Soc Edin, 1992, 120A: 185–189 41. Šverák V . Newexamples of quasiconvex functions. Arch Rational Mech Anal, 1992, 119: 293–300. doi:10.1007/BF01837111 42. Šverák V . Onthe problem of two wells. In: Microstructure and Phase Transition, IMA Vol Math Appl, 54. 1994, 183–189 43. Sychev M A . Comparing two methods of resolving homogeneous differential inclusions. Calc Var PDEs, 2001, 13: 213–229. doi:10.1007/PL00009929 44. Tartar L . Compensatedcompactness and applications to partial differential equations. In: Knops R J, ed. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV. London: PitmanPress, 1979, 136–212 45. Tartar L . Someremarks on separately convex functions. Microstructure and phase transition. IMA Vol Math Appl, 54. New York: Springer, 1993, 191–204 46. Zhang K -W . A construction of quasiconvex functions with linear growth at infinity. Ann Sc Norm Sup Pisa, Serie IV, 1992, XIX: 313–326 47. Zhang K -W . On connected subsets of M^2×2 without rank-one connections. Proc Royal Soc Edin, 1997, 127A: 207–216 48. Zhang K -W . Quasiconvex functions, SO(n) andtwo elastic wells. Anal Nonlin H Poincaré, 1997, 14: 759–785. doi:10.1016/S0294-1449(97)80132-1 49. Zhang K -W . On the structure of quasiconvex hulls. Ann Inst H Poincaré-Analyse Nonlineaire, 1998, 15: 663–686. doi:10.1016/S0294-1449(99)80001-8 50. Zhang K -W . Maximal extension for linear spaces of real matrices with large rank. Proc Royal Soc Edin, 2001, 131A: 1481–1491. doi:10.1017/S0308210500001499 51. Zhang K -W . Estimates of quasicovnex polytopes in the calculus of variations. J Convex Anal, 2006, 13: 37–50 52. Zhang K -W . On coercivity and regularity for linear elliptic systems. Preprint, 2008