Abstract

We study the asymptotic behaviour of the solutions of some minimization problems for integral functionals with convex integrands, in two-dimensional domains with cracks, under perturbations of the cracks in the Hausdorff metric. In the first part of the paper, we examine conditions for the stability of the minimum problem via duality arguments in convex optimization. In the second part, we study the limit problem in some special cases when there is no stability, using the tool of Γ\Gamma-convergence.

Author information

Contact details are reproduced from the original publication and may be historical.

François Ebobisse

S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy
and: Dept. of Mathematics, University of Cape Town, Rondebosch 7700, South Africa

ebobisse@maths.uct.ac.za

Marcello Ponsiglione

S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy

ponsigli@sissa.it

Keywords

  • capacity
  • convex optimization
  • Gamma-convergence
  • Hausdorff distance
  • integral functional

Mathematics Subject Classification

31A15, 46N10, 49M29, 49J45

References

  1. L. Boccardo, F. Murat: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. TMA 19 (1992) 581–597.https://doi.org/10.1016/0362-546X(92)90023-8Google Scholar
  2. D. Bucur, N. Varchon: Boundary variation for a Neumann problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000) 807–821.Google Scholar
  3. D. Bucur, N. Varchon: A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal. 34(2) (2002) 460–477.https://doi.org/10.1137/S0036141002389579Google Scholar
  4. D. Bucur, P. Trebeschi: Shape optimization problem governed by a nonlinear state equation, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 949–963.https://doi.org/10.1017/S0308210500030006Google Scholar
  5. D. Bucur, J.-P. Zolesio: Continuité par rapport au domaine dans le problème de Neumann, C. R. Acad. Sci. Paris, Sér. I 319 (1994) 57–60.Google Scholar
  6. D. Bucur, J.-P. Zolesio: Shape continuity for Dirichlet-Neumann problems, M. Chipot et al. (eds.), Progress in Partial Differential Equations: the Metz Surveys 4. (Metz, 1994-95), Pitman Res. Notes in Math. 345, Longman, Harlow (1996) 53–65.Google Scholar
  7. G. Buttazzo, G. Dal Maso: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim. 23 (1991) 17–49.https://doi.org/10.1007/BF01442391Google Scholar
  8. A. Chambolle, F. Doveri: Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Comm. Partial Differential Equations 22 (1997) 811–840.https://doi.org/10.1080/03605309708821285Google Scholar
  9. D. Chenais: On the existence of a solution in domain identification problem, J. Math Anal. Appl. 52 (1975) 189–289.https://doi.org/10.1016/0022-247X(75)90091-8Google Scholar
  10. G. Cortesani: Asymptotic behaviour of a sequence of Neumann problems, Comm. Partial Differential Equations 22 (1997) 1691–1729.https://doi.org/10.1080/03605309708821316Google Scholar
  11. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  12. G. Dal Maso, F. Ebobisse, M. Ponsiglione: A stability result for nonlinear Neumann problems under boundary variations, Preprint SISSA (June 2002), to appear in J. Math. Pures Appl. (9)https://doi.org/10.1016/S0021-7824(03)00014-XGoogle Scholar
  13. G. Dal Maso, U. Mosco: Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986) 345–387.https://doi.org/10.1007/BF00276841Google Scholar
  14. G. Dal Maso, F. Murat: Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XXIV (1997) 239–290.Google Scholar
  15. G. Dal Maso, R. Toader: A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002) 101–135.https://doi.org/10.1007/s002050100187Google Scholar
  16. G. Dal Maso, R. Toader: A model for the quasi-static growth of brittle fractures based on local minimization, Preprint SISSA (June 2002).https://doi.org/10.1142/S0218202502002331Google Scholar
  17. A. Damlamian: Le problème de la passoire de Neumann, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985) 427–450.Google Scholar
  18. J. Deny, J. L. Lions: Les espaces du type Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953) 305–370.https://doi.org/10.5802/aif.55Google Scholar
  19. F. Ebobisse, M. Ponsiglione: Stability of some unilateral free-discontinuity problems in two-dimensional domains, Preprint SISSA (July 2002).https://doi.org/10.1017/S0308210500002791Google Scholar
  20. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, Amsterdam, North-Holland (1976).Google Scholar
  21. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).Google Scholar
  22. K. J. Falconer: The Geometry of Fractal Sets, Cambridge University Press, Cambridge (1985).https://doi.org/10.1017/CBO9780511623738Google Scholar
  23. G. A. Francfort, J.-J. Marigo: Revisiting brittle fracture as energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342.https://doi.org/10.1016/S0022-5096(98)00034-9Google Scholar
  24. J. Heinonen, T. Kipelanen, O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford (1993).Google Scholar
  25. V. G. Maz’ya: Sobolev Spaces, Springer-Verlag, Berlin (1985).https://doi.org/10.1007/978-3-662-09922-3Google Scholar
  26. F. Murat: The Neumann sieve, Nonlinear Variational Problems (Isola d’Elba, 1983), A. Marino et al. (eds.), Res. Notes in Math. 127, Pitman, London (1985) 24–32.Google Scholar
  27. W. Ziemer: Weakly Differentiable Functions, Springer-Verlag, Berlin (1989).https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

F. Ebobisse, M. Ponsiglione. “A Duality Approach for Variational Problems in Domains with Cracks.” Journal of Convex Analysis 11 (2004), No. 1, 17–40. https://doi.org/10.68381/jca11002

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.