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Direct approach to the problem of strong local minima in calculus of variations

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An Erratum to this article was published on 14 February 2008

Abstract

The paper introduces a general strategy for identifying strong local minimizers of variational functionals. It is based on the idea that any variation of the integral functional can be evaluated directly in terms of the appropriate parameterized measures. We demonstrate our approach on a problem of W 1,∞ sequential weak-* local minima — a slight weakening of the classical notion of strong local minima. We obtain the first quasiconvexity-based set of sufficient conditions for W 1,∞ sequential weak-* local minima.

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References

  1. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(4), 337–403, (1976/1977)

  2. Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988), vol. 344 of Lecture Notes in Phys., pp. 207–215. Springer, Berlin Heidelberg New York (1989)

  3. Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, Berlin Heidelberg New York (2002)

  4. Ball J.M., Marsden J.E. (1984) Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal. 86(3):251–277

    Article  MATH  MathSciNet  Google Scholar 

  5. Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107(3), 655–663 (1989)

    Google Scholar 

  6. Carathéodory C. (1929) Über die Variationsrechnung bei mehrfachen Integralen. Acta Math. Szeged 4:401–426

    Google Scholar 

  7. Dacorogna B. (1982) Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal. 46(1):102–118

    Article  MATH  MathSciNet  Google Scholar 

  8. DeDonder, T.: Théorie invariantive du clacul des variations. Hayez, Brussels (1935)

  9. Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, (1992)

  10. Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, Berlin Heidelberg New York (1969)

  11. Fonseca I. (1992) Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh Sect. A 120(1–2):99–115

    MathSciNet  Google Scholar 

  12. Fonseca, I. Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29(3), 736–756 (electronic) (1998)

    Google Scholar 

  13. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin Heidelberg New York (2001) Reprint of the 1998 edition

  14. Grabovsky, Y., Truskinovsky, L.: Metastability in nonlinear elsticity. To be submitted

  15. Hestenes M.R. (1948) Sufficient conditions for multiple integral problems in the calculus of variations. Am. J. Math. 70:239–276

    Article  MATH  MathSciNet  Google Scholar 

  16. Hüsseinov F. (1995) Weierstrass condition for the general basic variational problem. Proc. Roy. Soc. Edinburgh Sect. A 125(4):801–806

    MATH  MathSciNet  Google Scholar 

  17. Kinderlehrer D., Pedregal P. (1991) Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115(4):329–365

    Article  MATH  MathSciNet  Google Scholar 

  18. Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4(1), 59–90 (1994)

    Google Scholar 

  19. Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39, 113–137, 139–182 and 353–377 (1986)

    Google Scholar 

  20. Kristensen, J.: Finite functionals and young measures generated by gradients of sobolev functions. Technical Report Mat-Report No. 1994-34, Mathematical Institute, Technical University of Denmark (1994)

  21. Kristensen J., Taheri A. (2003) Partial regularity of strong local minimizers in the multi-dimensional calculus of variations. Arch. Ration. Mech. Anal. 170(1):63–89

    Article  MATH  MathSciNet  Google Scholar 

  22. Lepage, T.: Sur les champs géodésiques des intégrales multiples. Acad. Roy. Belgique. Bull. Cl. Sci. (5) 27, 27–46 (1941)

    Google Scholar 

  23. Meyers N.G. (1965) Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Am. Math. Soc. 119:125–149

    Article  MATH  MathSciNet  Google Scholar 

  24. Morrey, C.B. Jr.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2, 25–53 (1952)

    Google Scholar 

  25. Müller, S., Šverák, V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. (2) 157(3), 715–742 (2003)

    Google Scholar 

  26. Pedregal, P.: Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications, vol. 30. Birkhäuser Verlag, Basel (1997)

  27. Post K.D.E., Sivaloganathan J. (1997) On homotopy conditions and the existence of multiple equilibria in finite elasticity. Proc. Roy. Soc. Edinburgh Sect. A 127(3):595–614

    MATH  MathSciNet  Google Scholar 

  28. Rudin, W.: Real and Complex Analysis 3rd edn. McGraw-Hill, New York (1987)

  29. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, (1970)

  30. Székelyhidi L. Jr. (2004) The regularity of critical points of polyconvex functionals. Arch. Ration. Mech. Anal. 172(1):133–152

    Article  MATH  MathSciNet  Google Scholar 

  31. Taheri, A.: Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations. Proc. Roy. Soc. Edinburgh Sect. A 131(1), 155–184 (2001)

  32. Taheri, A.: Local minimizers and quasiconvexity—the impact of topology. Arch. Ration. Mech. Anal. 176(3), 363–414 (2005)

    Google Scholar 

  33. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math. pp. 136–212. Pitman, Boston (1979)

  34. Tonelli, L.: Fondamenti di Calcolo delle Variazioni, vol. I, II. Nicola Zanichelli, Bologna (1921, 1923)

  35. Weyl H. (1935) Geodesic fields in the calculus of variations of multiple integrals. Ann. Math. 36:607–629

    Article  MathSciNet  Google Scholar 

  36. Young, L.C.: Approximation by polygons in the calculus of variations. Proc. Roy. Soc. London, Ser. A 141, 325–341 (1933)

    Google Scholar 

  37. Young, L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendue, Soc. Sci. Lett. Warsaw, cl. III 30, 212–234 (1933)

  38. Zhang K. (1992) Remarks on quasiconvexity and stability of equilibria for variational integrals. Proc. Am. Math. Soc. 114(4):927–930

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Temple University, Philadelphia, PA, 19122, USA

    Yury Grabovsky & Tadele Mengesha

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  1. Yury Grabovsky
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  2. Tadele Mengesha
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Correspondence to Yury Grabovsky.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s00526-007-0157-y

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Grabovsky, Y., Mengesha, T. Direct approach to the problem of strong local minima in calculus of variations. Calc. Var. 29, 59–83 (2007). https://doi.org/10.1007/s00526-006-0056-7

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