Advertisement

Manuscripta Mathematica

, Volume 140, Issue 1–2, pp 83–114 | Cite as

BMO and uniform estimates for multi-well problems

  • Georg Dolzmann
  • Jan KristensenEmail author
  • Kewei Zhang
Article

Abstract

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact set K in matrix space \({{{\mathbb M}^{N \times n}}}\). The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably convex, where we show that extremals are locally Lipschitz continuous. The results rely on the special structure of the integrand and elementary Cordes–Nirenberg type estimates for elliptic systems.

Mathematics Subject Classification (1991)

49Q20 35B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Fusco N.: Local regularity for minimizers of quasiconvex integrals. Ann. Sc. Norm. Sup. Pisa Cl. Sci. IV. Ser. 16, 603–636 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Asplund E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ball M.J., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100(1), 13–52 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ball M.J., James D.R.: Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. R. Soc. Lond. A 338, 389–450 (1992)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ball M.J., Kirchheim B., Kristensen J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11(4), 333–359 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Campanato S.: Equazioni ellittiche del \({{\rm II}^\circ}\) ordine espazi \({{\mathcal{L}}^{(2,\lambda)}}\). Ann. Mat. Pura Appl. 69(4), 321–381 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Carozza, M., Passarelli di Napoli, A., Schmidt, T., Verde, A.: Local and asymptotic regularity results for quasiconvex and quasimonotone problems. Quart. J. Math. (2012), to appearGoogle Scholar
  8. 8.
    Chipot M., Evans C.L.: Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations. Proc. R. Soc. Edinburgh Sect. A 102(3–4), 291–303 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chipot M., Kinderlehrer D.: Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103(3), 237–277 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dacorogna B.: Direct Methods in the Calculus of Variations, 2nd edn., vol. 78 of Applied Mathematical Sciences. Springer, New York (2008)Google Scholar
  11. 11.
    de Leeuw K., Mirkil H.: Majorations dans L des opérateurs différentiels à coefficients constants. C. R. Acad. Sci. Paris 254, 2286–2288 (1962)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dolzmann G., Kirchheim B., Kristensen J.: Conditions for equality of hulls in the calculus of variations. Arch. Ration. Mech. Anal. 154(2), 93–100 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dolzmann G., Kristensen J.: Higher integrability of minimizing Young measures. Calc. Var. Partial Differ. Equ. 22(3), 283–301 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ekeland, I., Témam, R. : Convex Analysis and Variational Problems, English edn., vol. 28 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999), translated from the FrenchGoogle Scholar
  15. 15.
    Evans C.L., Gariepy F.R.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  16. 16.
    Fonseca, I., Fusco, N.: Regularity results for anisotropic image segmentation models. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 24(3), 463–499 (1997)Google Scholar
  17. 17.
    Fuchs M.: Differentiability properties of minima of nonsmooth variational integrals. Ricerche Mat. 46(1), 23–29 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Iwaniec T.: The best constant in a BMO-inequality for the Beurling–Ahlfors transform. Michigan Math. J. 33(3), 387–394 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Iwaniec, T. : p-Harmonic tensors and quasiregular mappings. Ann. Math. 136(3), 589–624 (1992)Google Scholar
  20. 20.
    Kohn V.R.: The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3(3), 193–236 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kristensen J., Mingione G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180(3), 331–398 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kristensen J., Mingione G.: Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198(2), 369–455 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, vol. 51 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)Google Scholar
  24. 24.
    Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Müller, S.: Variational models for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), vol. 1713 of Lecture Notes in Math., pp. 85–210. Springer, Berlin (1999)Google Scholar
  26. 26.
    Rockafellar, R. T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Reprint of the 1970 original, Princeton PaperbacksGoogle Scholar
  27. 27.
    Scheven, C., Schmidt, T.: Asymptotically regular problems. II. Partial Lipschitz continuity and a singular set of positive measure. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8(3), 469–507 (2009)Google Scholar
  28. 28.
    Scheven C., Schmidt T.: Asymptotically regular problems. I. Higher integrability. J. Differ. Equ. 248(4), 745–791 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Seregin G.A.: On the regularity of solutions of variational problems in the theory of phase transitions in an elastic body. Algebra i Analiz 7(6), 153–187 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Seregin G.A.: A variational problem on the phase equilibrium of an elastic body. Algebra i Analiz 10(3), 92–132 (1998)MathSciNetGoogle Scholar
  31. 31.
    Šilhavý, M.: Rank 1 convex hulls of isotropic functions in dimension 2 by 2. In: Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), vol. 126, pp. 521–529 (2001)Google Scholar
  32. 32.
    Šverák V., Yan X.: Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl Acad. Sci. USA 99(24), 15269–15276 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Zhang K.: On various semiconvex relaxations of the squared-distance function. Proc. R. Soc. Edinburgh Sect. A 129(6), 1309–1323 (1999)zbMATHCrossRefGoogle Scholar
  34. 34.
    Zhang K.: A two-well structure and intrinsic mountain pass points. Calc. Var. Partial Differ. Equ. 13(2), 231–264 (2001)zbMATHCrossRefGoogle Scholar
  35. 35.
    Zhang K.: An elementary derivation of the generalized Kohn-Strang relaxation formulae. J. Convex Anal. 9(1), 269–285 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsSwansea UniversitySwanseaUK

Personalised recommendations