Skip to main content

A compactness result in \(GSBV^p\) and applications to \(\varGamma \)-convergence for free discontinuity problems

Abstract

We present a compactness result in the space \(GSBV^p\) which extends the classical statement due to Ambrosio (Arch Ration Mech 111:291–322, 1990) to problems without a priori bounds on the functions. As an application, we revisit the \(\varGamma \)-convergence results for free discontinuity functionals established recently by Cagnetti et al. [Ann Inst H Poincaré Anal Non Linéaire (to appear)]. We investigate sequences of boundary value problems and show convergence of minimum values and minimizers.

This is a preview of subscription content, access via your institution.

References

  1. Ambrosio, L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3, 857–881 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111, 291–322 (1990)

    MathSciNet  Article  Google Scholar 

  3. Ambrosio, L.: On the lower semicontinuity of quasi-convex integrals in \(SBV(\Omega; {\mathbb{R}}^k)\). Nonlinear Anal. 23, 405–425 (1994)

    MathSciNet  Article  Google Scholar 

  4. Ambrosio, L., Braides, A.: Functionals defined on partitions of sets of finite perimeter, II: semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, 307–333 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  6. Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6, 105–123 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Babadjian, J.F., Giacomini, A.: Existence of strong solutions for quasi-static evolution in brittle fracture. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13, 925–974 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Barenblatt, G.I.: The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Advances in Applied Mechanics, vol. 7, pp. 55–129. Academic Press, New York (1962)

    Google Scholar 

  9. Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in \(SBD(\Omega )\). Math. Z. 228, 337–351 (1998)

    MathSciNet  Article  Google Scholar 

  10. Braides, A.: \(\Gamma \)-Convergence for Beginners. Oxford University Press, Oxford (2002)

    Book  Google Scholar 

  11. Braides, A., Defranceschi, A., Vitali, E.: Homogenization of free discontinuity problems. Arch. Ration. Mech. Anal. 135, 297–356 (1996)

    MathSciNet  Article  Google Scholar 

  12. Cagnetti, F., Dal Maso, G., Scardia, L., Zeppieri, C.I.: \(\Gamma \)-convergence of free-discontinuity problems. Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. http://cvgmt.sns.it/paper/3371/

  13. Cagnetti, F., Dal Maso, G., Scardia, L., Zeppieri, C.I.: Stochastic Homogenisation of Free-Discontinuity Problems. Preprint (2017). http://cvgmt.sns.it/paper/3708/

  14. Cagnetti, F., Scardia, L.: An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains. J. Math. Pures Appl. 95, 349–381 (2011)

    MathSciNet  Article  Google Scholar 

  15. Cagnetti, F., Toader, R.: Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach. ESAIM Control Optim. Calc. Var. 17, 1–27 (2011)

    MathSciNet  Article  Google Scholar 

  16. Chambolle, A.: A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167, 167–211 (2003)

    MathSciNet  Article  Google Scholar 

  17. Chambolle, A., Conti, S., Francfort, G.: Korn–Poincaré inequalities for functions with a small jump set. Indiana Univ. Math. J. 65, 1373–1399 (2016)

    MathSciNet  Article  Google Scholar 

  18. Chambolle, A., Conti, S., Iurlano, F.: Approximation of functions with small jump sets and existence of strong minimizers of Griffith’s energy. Preprint (2017). arXiv:1710.01929

  19. Chambolle, A., Crismale, V.: A density result in \(GSBD^p\) with applications to the approximation of brittle fracture energies. Preprint (2017). Available at: arXiv:1708.03281

  20. Chambolle, A., Crismale, V.: Compactness and lower semicontinuity in \(GSBD\). J. Eur. Math. Soc. (JEMS), to appear. http://cvgmt.sns.it/paper/3767/

  21. Conti, S., Focardi, M., Iurlano, F.: Existence of strong minimizers for the Griffith static fracture model in dimension two. Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. arXiv:1611.03374

  22. Cortesani, G.: Strong approximation of GSBV functions by piecewise smooth functions. Ann. Univ. Ferrara Sez. 43, 27–49 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Crismale, V., Lazzaroni, G., Orlando, G.: Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue. Math. Models Methods Appl. Sci. 28, 1371–1412 (2018). https://doi.org/10.1142/S0218202518500379

    MathSciNet  Article  MATH  Google Scholar 

  24. Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Boston (1993)

    Book  Google Scholar 

  25. Dal Maso, G.: Generalized functions of bounded deformation. J. Eur. Math. Soc. (JEMS) 15, 1943–1997 (2013)

    MathSciNet  Article  Google Scholar 

  26. Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)

    MathSciNet  Article  Google Scholar 

  27. Dal Maso, G., Giacomini, A., Ponsiglione, M.: A variational model for quasistatic crack growth in nonlinear elasticity: qualitative properties of the solutions. Boll. Unione Mat. Ital. 2, 371–390 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Dal Maso, G., Lazzaroni, G.: Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 257–290 (2010)

    MathSciNet  Article  Google Scholar 

  29. Dal Maso, G., Zanini, C.: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. R. Soc. Edinb. Sect. A 137, 253–279 (2007)

    MathSciNet  Article  Google Scholar 

  30. Francfort, G.A., Larsen, C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56, 1465–1500 (2003)

    MathSciNet  Article  Google Scholar 

  31. Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    MathSciNet  Article  Google Scholar 

  32. Friedrich, M.: A derivation of linearized Griffith energies from nonlinear models. Arch. Ration. Mech. Anal. 225, 425–467 (2017)

    MathSciNet  Article  Google Scholar 

  33. Friedrich, M.: A piecewise Korn inequality in SBD and applications to embedding and density results. SIAM J. Math. Anal. 50, 3842–3918 (2018)

    MathSciNet  Article  Google Scholar 

  34. Friedrich, M., Schmidt, B.: On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime. Netw. Heterog. Media 10, 321–342 (2015)

    MathSciNet  Article  Google Scholar 

  35. Friedrich, M., Solombrino, F.: Quasistatic crack growth in \(2d\)-linearized elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 27–64 (2018)

    MathSciNet  Article  Google Scholar 

  36. Giacomini, A., Ponsiglione, M.: A \(\Gamma \)-convergence approach to stability of unilateral minimality properties. Arch. Ration. Mech. Anal. 180, 399–447 (2006)

    MathSciNet  Article  Google Scholar 

  37. Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221, 163–198 (1921)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Manuel Friedrich.

Additional information

Publisher's Note

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

Communicated by L. Ambrosio.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Friedrich, M. A compactness result in \(GSBV^p\) and applications to \(\varGamma \)-convergence for free discontinuity problems. Calc. Var. 58, 86 (2019). https://doi.org/10.1007/s00526-019-1530-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1530-3

Mathematics Subject Classification

  • 49J45
  • 49Q20
  • 70G75
  • 74Q05
  • 74R10