Short Tales from Nonlinear Calderón-Zygmund Theory

Part of the Lecture Notes in Mathematics book series (LNM, volume 2186)


Nonlinear Calderón-Zygmund Theory aims at reproducing, in the nonlinear setting, the classical linear theory originally developed by Calderón and Zygmund. This topic has large intersections with Nonlinear Potential Theory. We survey here the main results of this theory.


  1. 1.
    E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. (Crelles J.) 584, 117–148 (2005)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314 (Springer, Berlin, 1996)Google Scholar
  4. 4.
    S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    B. Avelin, T. Kuusi, G. Mingione, Nonlinear Calderón-Zygmund theory in the limiting case. Arch. Ration Mech. Anal. DOI: 10.1007/s00205-017-1171-7Google Scholar
  6. 6.
    L. Beck, G. Mingione, Lipschitz bounds and non-uniform ellipticity (to appear)Google Scholar
  7. 7.
    P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems. J. Differ. Equ. 255, 2927–2951 (2013)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    P. Baroni, Marcinkiewicz estimates for degenerate parabolic equations with measure data. J. Funct. Anal. 267, 3397–3426 (2014)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    P. Baroni, Riesz potential estimates for a general class of quasilinear equations. Calc. Var. Partial Differ. Equ. 53, 803–846 (2015)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    P. Baroni, J. Habermann, Calderón-Zygmund estimates for parabolic measure data equations. J. Differ. Equ. 252, 412–447 (2012)MATHCrossRefGoogle Scholar
  11. 11.
    P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vázquez, An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (IV) 22, 241–273 (1995)MathSciNetMATHGoogle Scholar
  12. 12.
    L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87,149–169 (1989)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    L. Boccardo, T. Gallouët, J.L. Vázquez, Nonlinear elliptic equations in \(\mathbb{R}^{N}\) without growth restrictions on the data. J. Differ. Equ. 105, 334–363 (1993)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincarè Anal. Non Linéaire 13, 539–551 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258 (1997)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    V. Bögelein, F. Duzaar, G. Mingione, Degenerate problems with irregular obstacles. J. Reine Angew. Math. (Crelles J.) 650, 107–160 (2011)MathSciNetMATHGoogle Scholar
  18. 18.
    L. Brasco, F. Santambrogio, A sharp estimate à Calderón-Zygmund for the p-Laplacian. Commun. Contemp. Math. (to appear)Google Scholar
  19. 19.
    D. Breit, A. Cianchi, L. Diening, T. Kuusi, S. Schwarzacher, Pointwise Calderón-Zygmund gradient estimates for the p-Laplace system. J. Math. Pures Appl.
  20. 20.
    H. Brezis, W.A. Strauss, Semi-linear second-order elliptic equations in L 1. J. Math. Soc. Japan 25, 565–590 (1973)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    S.S. Byun, Elliptic equations with BMO coefficients in Lipschitz domains. Trans. Am. Math. Soc. 357, 1025–1046 (2005)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    S.S. Byun, Parabolic equations with BMO coefficients in Lipschitz domains. J. Differ. Equ. 209, 229–265 (2005)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    S. Byun, Y. Cho, L. Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles. J. Funct. Anal. 263, 3117–3143 (2012)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (II) 130, 189–213 (1989)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    L. Caffarelli, I. Peral, On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    A.P. Calderón, A. Zygmund, On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    A.P. Calderón, A. Zygmund, On singular integrals. Am. J. Math. 78, 289–309 (1956)MATHCrossRefGoogle Scholar
  28. 28.
    S. Campanato, G. Stampacchia, Sulle maggiorazioni in L p nella teoria delle equazioni ellittiche. Boll. Un. Mat. Ital. (III) 20, 393–399 (1965)MathSciNetMATHGoogle Scholar
  29. 29.
    A. Cianchi, Nonlinear potentials, local solutions to elliptic equations and rearrangements. Ann. Scu. Norm. Sup. Cl. Sci. (V) 10, 335–361 (2011)MathSciNetMATHGoogle Scholar
  30. 30.
    A. Cianchi, V. Maz’ya, Global Lipschitz regularity for a class of quasilinear equations. Commun. Partial Differ. Equ. 36, 100–133 (2011)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    A. Cianchi, V. Maz’ya, Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch. Ration. Mech. Anal. 212, 129–177 (2014)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    A. Cianchi, V. Maz’ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems. J. Eur. Math. Soc. 16, 571–595 (2014)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    M. Colombo, G. Mingione, Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    M. Colombo, G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    A. Coscia, G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris Sér. I Math. 328, 363–368 (1999)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (IV) 28, 741–808 (1999)MathSciNetMATHGoogle Scholar
  38. 38.
    T. Daskalopoulos, T. Kuusi, G. Mingione, Borderline estimates for fully nonlinear elliptic equations. Commun. Partial Differ. Equ. 39, 574–590 (2014)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III) 125(3), 25–43 (1957)Google Scholar
  40. 40.
    E. De Giorgi, Frontiere Orientate di Misura Minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–1961 (Editrice Tecnico Scientifica, Pisa, 1961), p. 57Google Scholar
  41. 41.
    R.A. DeVore, R.C. Sharpley, Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47(293) (1984)Google Scholar
  42. 42.
    E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    E. DiBenedetto, Degenerate Parabolic Equations. Universitext (Springer, New York, 1993)MATHCrossRefGoogle Scholar
  44. 44.
    E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115, 1107–1134 (1993)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    G. Dolzmann, N. Hungerbühler, S. Müller, The p-harmonic system with measure-valued right-hand side. Ann. Inst. H. Poincarè Anal. Non Linèaire 14, 353–364 (1997)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    G. Dolzmann, N. Hungerbühler, S. Müller, Nonlinear elliptic systems with measure-valued right-hand side. Math. Z. 226, 545–574 (1997)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    G. Dolzmann, N. Hungerbühler, S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side. J. Reine Angew. Math. (Crelles J.) 520, 1–35 (2000)MathSciNetMATHGoogle Scholar
  48. 48.
    F. Duzaar, G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Differ. Equ. 20, 235–256 (2004)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    F. Duzaar, G. Mingione, Regularity for degenerate elliptic problems via p-harmonic approximation. Ann. Inst. H. Poincaré Anal. Non Linèaire 21, 735–766 (2004)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    F. Duzaar, G. Mingione, Local Lipschitz regularity for degenerate elliptic systems. Ann. Inst. H. Poincarè Anal. Non Linèaire 27, 1361–1396 (2010)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259, 2961–2998 (2010)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    F. Duzaar, G. Mingione, Gradient continuity estimates. Calc. Var. Partial Differ. Equ. 39, 379–418 (2010)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    F. Duzaar, G. Mingione, Gradient estimates via nonlinear potentials. Am. J. Math. 133, 1093–1149 (2011)MATHCrossRefGoogle Scholar
  54. 54.
    F. Duzaar, J. Kristensen, G. Mingione, The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. (Crelles J.) 602, 17–58 (2007)MathSciNetMATHGoogle Scholar
  55. 55.
    F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth, and regularity. Mem. Am. Math. Soc. 214(1005), 128 (2011)Google Scholar
  56. 56.
    L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with ( p, q) growth. J. Differ. Equ. 204, 5–55 (2004)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    M. Fuchs, The blow-up of p-harmonic maps. Manuscripta Math. 81, 89–94 (1993)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    E. Giusti, Direct Methods in the Calculus of Variations (World Scientific, River Edge, NJ, 2003)MATHCrossRefGoogle Scholar
  59. 59.
    L. Greco, T. Iwaniec, C. Sbordone, Inverting the p-harmonic operator. Manuscripta Math. 92, 249–258 (1997)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    M. Havin, V.G. Maz’ja, Nonlinear potential theory. Russ. Math. Surv. 27, 71–148 (1972)MathSciNetGoogle Scholar
  61. 61.
    L. Hedberg, T.H. Wolff, Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33, 161–187 (1983)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs (Oxford University Press, New York, 1993)MATHGoogle Scholar
  63. 63.
    T. Iwaniec, Projections onto gradient fields and L p-estimates for degenerated elliptic operators. Stud. Math. 75, 293–312 (1983)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    T. Iwaniec, p-harmonic tensors and quasiregular mappings. Ann. Math. (II) 136, 589–624 (1992)Google Scholar
  65. 65.
    T. Iwaniec, C. Sbordone, Weak minima of variational integrals. J. Reine Angew. Math. (Crelles J.) 454, 143–161 (1994)MathSciNetMATHGoogle Scholar
  66. 66.
    F. John, L. Nirenberg, On functions of bounded mean oscillation. Commun. Pure Appl. Math.14, 415–426 (1961)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 19, 591–613 (1992)MathSciNetMATHGoogle Scholar
  68. 68.
    T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    T. Kilpeläinen, T. Kuusi, A. Tuhola-Kujanpää, Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincarè, Anal. Non Linèaire 28, 775–795 (2011)Google Scholar
  70. 70.
    J. Kinnunen, J.L. Lewis, Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102, 253–271 (2000)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    J. Kristensen, G. Mingione, The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    J. Kristensen, G. Mingione, Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198, 369–455 (2010)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    N.V. Krylov, Parabolic and elliptic equations with VMO-coefficients. Commun. Partial Differ. Equ. 32, 453–475 (2007)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    N.V. Krylov, Parabolic equations with VMO-coefficients in Sobolev spaces with mixed norms. J. Funct. Anal. 250, 521–558 (2007)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iber. 28, 535–576 (2012)MathSciNetMATHGoogle Scholar
  76. 76.
    T. Kuusi, G. Mingione, Universal potential estimates. J. Funct. Anal. 262, 4205–4638 (2012)MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    T. Kuusi, G. Mingione, New perturbation methods for nonlinear parabolic problems. J. Math. Pures Appl. (IX) 98, 390–427 (2012)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 215–246 (2013)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    T. Kuusi, G. Mingione, Gradient regularity for nonlinear parabolic equations. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 12, 755–822 (2013)MathSciNetMATHGoogle Scholar
  80. 80.
    T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations. Arch. Ration. Mech. Anal. 212, 727–780 (2014)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    T. Kuusi, G. Mingione, Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations. J. Eur. Math. Soc. 16, 835–892 (2014)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    T. Kuusi, G. Mingione, A nonlinear Stein theorem. Calc. Var. Partial Differ. Equ. 51, 45–86 (2014)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    T. Kuusi, G. Mingione, Borderline gradient continuity for nonlinear parabolic systems. Math. Ann. 360, 937–993 (2014)MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    T. Kuusi, G. Mingione, Partial regularity and potentials. J. École Pol. Math. 3, 309–363 (2016)MathSciNetMATHGoogle Scholar
  86. 86.
    T. Kuusi, G. Mingione, Vectorial nonlinear potential theory. J. Eur. Math. Soc. (to appear)Google Scholar
  87. 87.
    T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data. Commun. Math. Phys. 337, 1317–1368 (2015)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    O.A. Ladyzhenskaya, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York/London, 1968)MATHGoogle Scholar
  89. 89.
    J.L. Lewis, On very weak solutions of certain elliptic systems. Commun. Partial Differ. Equ. 18, 1515–1537 (1993)MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    P. Lindqvist, On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. (Crelles J.) 365, 67–79 (1986)MathSciNetMATHGoogle Scholar
  91. 91.
    P. Lindqvist, O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations. Acta Math. 155, 153–171 (1985)MATHGoogle Scholar
  92. 92.
    W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients. Ann. Scu. Norm. Sup. Pisa (III) 17, 43–77 (1963)MathSciNetMATHGoogle Scholar
  93. 93.
    J. Malý, W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, vol. 51 (American Mathematical Society, Providence, RI, 1997)Google Scholar
  94. 94.
    J.J. Manfredi, Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations, Ph.D. Thesis, University of Washington, St. LouisGoogle Scholar
  95. 95.
    P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105, 267–284 (1989)MATHCrossRefGoogle Scholar
  96. 96.
    P. Marcellini, Regularity and existence of solutions of elliptic equations with p-growth conditions. J. Differ. Equ. 90, 1–30 (1991)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    V. Maz’ya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations. (Russian) Vestnik Leningrad. Univ. 25, 42–55, (1970)Google Scholar
  98. 98.
    G. Mingione, The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166, 287–301 (2003)MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    G. Mingione, Regularity of minima: an invitation to the Dark side of the Calculus of Variations. Appl. Math. 51, 355–425 (2006)MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 6, 195–261 (2007)MATHGoogle Scholar
  101. 101.
    G. Mingione, Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    G. Mingione, Gradient potential estimates. J. Eur. Math. Soc. 13, 459–486 (2011)MathSciNetMATHGoogle Scholar
  103. 103.
    G. Mingione, Nonlinear measure data problems. Milan J. Math. 79, 429–496 (2011)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    G. Mingione, Nonlinear Calderón-Zygmund Theory. EMS, ETH Zürich series (to appear)Google Scholar
  105. 105.
    N.C. Phuc, I.E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type. Ann. of Math. (II) 168, 859–914 (2008)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    N.C. Phuc, I.E. Verbitsky, Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    J. Serrin, Pathological solutions of elliptic differential equations. Ann. Scu. Norm. Sup. Pisa (III) 18, 385–387 (1964)MathSciNetMATHGoogle Scholar
  108. 108.
    J. Simon, Régularité de solutions de problèmes nonlinèaires. C. R. Acad. Sci. Paris Sr. A-B 282, 1351–1354 (1976)MATHGoogle Scholar
  109. 109.
    G. Stampacchia, \(\mathcal{L}^{(\,p,\lambda )}\)-spaces and interpolation. Commun. Pure Appl. Math. 17, 293–306 (1964)Google Scholar
  110. 110.
    E.M. Stein, Editor’s note: the differentiability of functions in \(\mathbb{R}^{n}\). Ann. Math. (II) 113, 383–385 (1981)MathSciNetMATHGoogle Scholar
  111. 111.
    V. Šverák, X. Yan, Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99(24), 15269–15276 (2002)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    N.S. Trudinger, X.J. Wang, On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369–410 (2002)MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems. Acta Math. 138, 219–240 (1977)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    N.N. Ural’tseva, Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 184–222 (1968)MathSciNetMATHGoogle Scholar
  115. 115.
    L. Wang, A geometric approach to the Calderón-Zygmund estimates. Acta Math. Sin. (Engl. Ser.) 19, 381–396 (2003)MathSciNetMATHCrossRefGoogle Scholar
  116. 116.
    V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)MathSciNetGoogle Scholar
  117. 117.
    V.V., Zhikov, On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)Google Scholar
  118. 118.
    V.V. Zhikov, On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di ParmaParmaItaly

Personalised recommendations