Skip to main content
SpringerLink
Log in

Global Regularity for Minimizers of Some Anisotropic Variational Integrals

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We give regularity results for minimizers of two special cases of polyconvex functionals. Under some structural assumptions on the energy density, we prove that minimizers are either bounded, or have suitable integrability properties, by using the classical Stampacchia Lemma.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. De Giorgi, E.: Un esempio di estremali discontinue per un problema variazionale ditipo ellittico. [An example of discontinuous extremals for an elliptic type variational problem]. Boll. Un. Mat. Ital. 4, 135–137 (1968)

    Google Scholar 

  2. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Arch. Rational Mech. Anal. 51, 355–426 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Carozza, M., Passarelli, A.: Model problems from nonlinear elasticity: partial regularity results. ESAIM Control Opt. Calc. Var. 13, 120–134 (2007)

    Article  MathSciNet  Google Scholar 

  4. Fuchs, M., Seregin, G.: Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity. Algebra Anal. 6, 128–153 (1994)

    MATH  Google Scholar 

  5. Fusco, N., Hutchinson, J.E.: Partial regularity in problem motivated by nonlinear elasticity. SIAM J. Math. Anal. 22, 1516–1551 (1991)

    Article  MathSciNet  Google Scholar 

  6. Fuchs, M., Reuling, G.: Partial regularity for certain classes of polyconvex functionals related to non linear elasticity. Manuscr. Math. 87, 13–26 (1995)

    Article  Google Scholar 

  7. Fusco, N., Hutchinson, J.E.: Partial regularity and everywhere continuity for a model problem from non-linear elasticity. J. Austral. Math. Soc. Ser. A 57, 158–169 (1994)

    Article  MathSciNet  Google Scholar 

  8. Cupini, G., Leonetti, F., Mascolo, E.: Local boundedness for mininizers of some polyconvex integrals. Arch. Rat. Mech. Anal. 224, 269–289 (2017)

    Article  Google Scholar 

  9. Leonetti, F.: Pointwise estimates for a model problem in nonlinear elasticity. In: Forum Math. vol. 18, pp. 529–534 (2006)

  10. D’Ottavio, A., Leonetti, F., Musciano, C.: Maximum principle for vector-valued mappings minimizing variational integrals. Atti. Sem. Mat. Fis. Modena 46, 677–683 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Leonetti, F., Siepe, F.: Maximum principle for vector-valued minimizers of some intergral functionals. J. Convex Anal. 12, 267–278 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Leonetti, F., Siepe, F.: Bounds for vector valued minimizers of some integral functionals. Ricerche Mat. 54, 303–312 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Gao, H., Leonetti, F., Macri, M., Petricca, P.V.: Regularity for minimizers with positive Jacobian. Appl. Anal. 39, 496–507 (2020)

    Article  MathSciNet  Google Scholar 

  14. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

    Article  MathSciNet  Google Scholar 

  15. Ball J.M.: Constitutive inequalities and existence thorems in elastostatics. In: Nonlinear Analysis and Mechanics, Proceedings, Research Notes Pitman, London, vol. 17, pp. 13–25 (1977)

  16. Bauman, P., Owen, N., Phillps, D.: Maximum principles and a priori estimates for a class of problems from nonlinear elasticity. Ann. Inst. Henri Poincaré Anal. Nonlineaire 8, 119–157 (1991)

    Article  MathSciNet  Google Scholar 

  17. Müller, S.: Higher integrability of determinats and weak convergence in \(L^1\). J. Reine Angew. Math. 412, 20–34 (1990)

    MathSciNet  Google Scholar 

  18. Dacorogna B.: Direct methods in the calculus of variations. In: Applied Mathematical Sciences, vol. 78, 2nd edn, Springer, New York (2008)

  19. Carozza, M., Gao, H., Giova, R., Leonetti, F.: Local boundedness for minimizers of some polyconvex integrals. J. Optim. Theory Appl. 178, 699–725 (2018)

    Article  MathSciNet  Google Scholar 

  20. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)

    Book  Google Scholar 

  21. Giaquinta, M.: Growth condition and regularity, a counterexample. Manuscr. Math. 59, 245–248 (1987)

    Article  MathSciNet  Google Scholar 

  22. Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity results for functionals with \((p, q)\) growth. J. Differ. Equ. 204, 5–55 (2004)

    Article  Google Scholar 

  23. Hong, M.C.: Some remarks on the minimizers of variational integrals with non standard growth conditions. Boll. Un Mat. Ital. A6, 91–101 (1992)

    MATH  Google Scholar 

  24. Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal. 105, 267–284 (1989)

    Article  MathSciNet  Google Scholar 

  25. Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p-q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)

    Article  MathSciNet  Google Scholar 

  26. Acerbi, E., Fusco, N.: Partial regularity under anisotropic \((p, q)\) growth conditions. J. Differential Equations 107, 46–67 (1994)

    Article  MathSciNet  Google Scholar 

  27. Moscariello, G., Nania, L.: Hölder continuity of minimizers of functionals with non standard growth conditions. Ricerche Mat. 40, 259–273 (1991)

    MathSciNet  MATH  Google Scholar 

  28. Marcellini, P.: Growth conditions and regularity for weak solutions to nonlinear elliptic pdes. J. Math. Anal. Appl. (2020). https://doi.org/10.1016/j.jmaa.2020.124408

  29. Tang, Q.: Regularity of minimizers of non-isotropic integrals of the calculus of variations. Ann. Mat. Pura Appl. 164, 77–87 (1993)

    Article  MathSciNet  Google Scholar 

  30. Lions, J.L.: Quelques methodes de resolution des problemes aux limites non lineaires [Some methods of solving non-linear boundary problems]. Dunod-Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  31. Fusco, N., Sbordone, C.: Some remarks on the regularity of minima of anisotropic integrals. Commun. Partial Differ. Equ. 18, 153–167 (1993)

    Article  MathSciNet  Google Scholar 

  32. Leonetti, F., Siepe, F.: Global integrability for minimizers of anisotropic functionals. Manuscr. Math. 144, 91–98 (2014)

    Article  MathSciNet  Google Scholar 

  33. Boccado L., Croce G.: Elliptic Partial Differential Equations. De Gruyter Studies in Mathematics vol 55, De Gruyter, Berlin (2014)

  34. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  35. Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. [Inclusion theorems for anisotropic Sobolev spaces]. Ricerche Mat. 18, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

  36. Stampacchia G.: Équations elliptiques du second ordre à coefficients discontinus [Second order elliptic equations with discontinuous coefficients] Séminaire Jean Leray, 1963–1964 (3), 1–77 (1963)

  37. Gao, H., Leonetti, F., Wang, L.: Remarks on Stampacchia lemma. J. Math. Anal. Appl. 458, 112–122 (2018)

    Article  MathSciNet  Google Scholar 

  38. Gao, H., Di, Q., Ma, D.: Integrability for solutions to some anisotropic obstacle problems. Manuscr. Math. 146, 433–444 (2015)

    Article  MathSciNet  Google Scholar 

  39. Innamorati, A., Leonetti, F.: Global integrability for weak solutoins to some anisotropic elliptic equations. Nonlinear Anal. 113, 430–434 (2015)

    Article  MathSciNet  Google Scholar 

  40. Kovalevsky, A.A.: Integrability and boundedness of solutions to some anisotropic problems. J. Math. Anal. Appl. 432, 820–843 (2015)

    Article  MathSciNet  Google Scholar 

  41. Leonetti, F., Petricca, P.V.: Global summability for solutoins to some anisotropic elliptic systems. Adv. Differ. Equ. 20, 1165–1186 (2015)

    MATH  Google Scholar 

  42. Kovalevskii, A.A.: On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations. Izvestiya Math. 67, 881–894 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Gao thanks NSF of Hebei Province (A2019201120) and NSFC(12071021) for the support; Ren thanks NSF Hebei Province (A2018201285), Science and Technology Project of Hebei Education Department(QN2020145) and Research Funds of Hebei University(8012605) for the support. All authors would like to thank the anonymous referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Hebei University, Baoding, China

    Hongya Gao, Miaomiao Huang & Wei Ren

Authors
  1. Hongya Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miaomiao Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Ren.

Additional information

Communicated by Irena Lasiecka.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, H., Huang, M. & Ren, W. Global Regularity for Minimizers of Some Anisotropic Variational Integrals. J Optim Theory Appl 188, 523–546 (2021). https://doi.org/10.1007/s10957-020-01795-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01795-7

Keywords

Mathematics Subject Classification

Search

Navigation