Archive for Rational Mechanics and Analysis

, Volume 184, Issue 2, pp 341–369 | Cite as

The Singular Set of Lipschitzian Minima of Multiple Integrals

  • Jan Kristensen
  • Giuseppe MingioneEmail author


The singular set of any Lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its Hausdorff dimension is strictly smaller than the space dimension


Hausdorff Dimension Multiple Integral Carleson Measure Partial Regularity Nonlinear Elliptic System 
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  1. 1.
    Acerbi E., Fusco N. (1984) Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86: 125–145zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Acerbi E., Fusco N. (1987) A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal. 99: 261–281zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ball J. (1976/77) Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63: 337–403CrossRefGoogle Scholar
  4. 4.
    Bonk M., Heinonen J. (2004) Smooth quasiregular mappings with branching. Publ. Math. Inst. Hautes Études Sci. 100: 153–170zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chipot M., Evans L.C. (1986) Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations. Proc. Roy. Soc. Edinburgh Sect. A 102: 291–303MathSciNetzbMATHGoogle Scholar
  6. 6.
    David G., Semmes S (1996) On the singular sets of minimizers of the Mumford-Shah functional. J. Math. Pures Appl. (9) 75: 299–342zbMATHMathSciNetGoogle Scholar
  7. 7.
    De Giorgi E. (1968) Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. Ital. (4) 1: 135–137zbMATHMathSciNetGoogle Scholar
  8. 8.
    Dolcini A., Esposito L., Fusco N. (1996) C 0,α regularity of ω-minima. Boll. Un. Mat. Ital. Sez. A 10: 113–125zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dolzmann, G.: Variational Methods for Crystalline Microstructure—Analysis and Computation. Lecture Notes in Math., 1803. Springer, 2003Google Scholar
  10. 10.
    Dorronsoro J.R. (1985) A characterization of potential spaces. Proc. Amer. Math. Soc. 95: 21–31zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Duzaar F., Gastel A. (2002) Nonlinear elliptic systems with Dini continuous coefficients. Arch. Math. (Basel) 78: 58–73zbMATHMathSciNetGoogle Scholar
  12. 12.
    Duzaar F., Gastel A., Mingione G. (2004) Elliptic systems, singular sets and Dini continuity. Comm. Partial Differential Equations 29: 1215–1240zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Duzaar F., Kronz M. (2002) Regularity of ω-minimizers of quasi-convex variational integrals with polynomial growth. Differential Geom. Appl. 17: 139–152zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Evans L.C. (1986) Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95: 227–252zbMATHCrossRefGoogle Scholar
  15. 15.
    Foss, M.: Global regularity for almost minimizers of nonconvex variational problems. Preprint 2006Google Scholar
  16. 16.
    Giaquinta, M.: Direct methods for regularity in the calculus of variations. Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. VI (Paris, 1982/1983), 258–274, Res. Notes in Math., 109, Pitman, Boston, MA, 1984Google Scholar
  17. 17.
    Giaquinta, M.: The problem of the regularity of minimizers. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1072–1083, Amer. Math. Soc., Providence, RI, 1987Google Scholar
  18. 18.
    Giaquinta, M.: Quasiconvexity, growth conditions and partial regularity. Partial Differential Equations and Calculus of Variations. Lecture Notes in Math., 1357, pp. 211–237, Springer, Berlin, 1988Google Scholar
  19. 19.
    Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993Google Scholar
  20. 20.
    Giusti E. (2003) Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge, NJzbMATHGoogle Scholar
  21. 21.
    Hamburger C. (1995) Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Mat. Pura Appl. (4) 169: 321–354zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems I, II, III. Comm. Pure Appl. Math. 39, 113–137, 139–182: 353–377 (1986)Google Scholar
  23. 23.
    Kristensen J., Mingione G. (2005) The singular set of ω-minima. Arch. Ration. Mech. Anal. 177: 93–114zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kristensen J., Mingione G. (2006) The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180: 331–398zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kristensen J., Taheri A. (2003) Partial regularity of strong local minimizers in the multi-dimensional calculus of variations. Arch. Ration. Mech. Anal. 170: 63–89zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Landes R. (1996) Quasimonotone versus pseudomonotone. Proc. Roy. Soc. Edinburgh. Sect. A 126: 705–717zbMATHMathSciNetGoogle Scholar
  27. 27.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995Google Scholar
  28. 28.
    Mingione G. (2003) The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166: 287–301zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mingione G. (2003) Bounds for the singular set of solutions to non linear elliptic systems. Calc. Var. Partial Differential Equations 18: 373–400zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mingione G. (2006) Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51Google Scholar
  31. 31.
    Morrey C.B. (1952) Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2: 25–53zbMATHMathSciNetGoogle Scholar
  32. 32.
    Müller, S.: Variational models for microstructure and phase transitions. Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Math., 1713, pp.85–210. Springer, 1999Google Scholar
  33. 33.
    Müller S., Scaron;verák V. (2003) Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157: 715–742MathSciNetCrossRefGoogle Scholar
  34. 34.
    Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. Theory of Nonlinear Operators (Proc. Fourth Internat. Summer School, Acad. Sci., Berlin, 1975), pp. 197–206Google Scholar
  35. 35.
    Raymond J. P. (1991) Lipschitz regularity of solutions of some asymptotically convex problems. Proc. Roy. Soc. Edinburgh Sect. A 117: 59–73MathSciNetGoogle Scholar
  36. 36.
    Rigot S. (2000) Uniform partial regularity of quasi minimizers for the perimeter. Calc. Var. Partial Differential Equations 10: 389–406zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Salli A. (1991) On the Minkowski dimension of strongly porous fractal sets in R n. Proc. London Math. Soc. (3) 62: 353–372zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Šverák V., Yan X. (2002) Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99: 15269–15276CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Székelyhidi L., Jr. (2004) The regularity of critical points of polyconvex functionals. Arch. Ration. Mech. Anal. 172: 133–152zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Zhang, K.: On the Dirichlet problem for a class of quasilinear elliptic systems of PDEs in divergence form. Partial Differential Equations, Proc. Tranjin 1986 (Ed. S.S. Chern). Lecture Notes in Math., 1306, pp. 262–277. Springer, 1988Google Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Dipartimento di MatematicaUniversità di ParmaParmaItaly

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