A duality approach for variational problems in domains with cracks

Journal of Convex Analysis

Volumn 11, Issue 1, 2004, Pages 17-40

  Ebobisse, François ^a,b   Ponsiglione, Marcello ^a  

^a SISSA   (Italy)

^b UNIVERSITY OF CAPE TOWN   (South Africa)

Author keywords

convergence; Capacity; Convex optimization; Hausdorff distance; Integral functional

Indexed keywords

EID: 7544228238     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

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.
2. D. Bucur, N. Varchon: Boundary variation for a Neumann problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000) 807-821.
3. D. Bucur, N. Varchon: A duality approach for the boundary variation of Neumann problems, SIAM J. Math. Anal. 34(2) (2002) 460-477.
4. D. Bucur, P. Trebeschi: Shape optimization problem governed by a nonlinear state equation, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 949-963.
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.
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.
7. G. Buttazzo, G. Dal Maso: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim. 23 (1991) 17-49.
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.
9. D. Chenais: On the existence of a solution in domain identification problem, J. Math Anal. Appl. 52 (1975) 189-289.
10. G. Cortesani: Asymptotic behaviour of a sequence of Neumann problems, Comm. Partial Differential Equations 22 (1997) 1691-1729.
11. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).
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)
13. G. Dal Maso, U. Mosco: Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986) 345-387.
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.
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.
16. G. Dal Maso, R. Toader: A model for the quasi-static growth of brittle fractures based on local minimization, Preprint SISSA (June 2002).
17. A. Damlamian: Le problème de la passoire de Neumann, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985) 427-450.
18. J. Deny, J. L. Lions: Les espaces du type Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953) 305-370.
19. F. Ebobisse, M. Ponsiglione: Stability of some unilateral free-discontinuity problems in two-dimensional domains, Preprint SISSA (July 2002).
20. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, Amsterdam, North-Holland (1976).
21. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).
22. K. J. Falconer. The Geometry of Fractal Sets, Cambridge University Press, Cambridge (1985).
23. G. A. Francfort, J.-J. Marigo: Revisiting brittle fracture as energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319-1342.
24. J. Heinonen, T. Kipelanen, O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford (1993).
25. V. G. Maz'ya: Sobolev Spaces, Springer-Verlag, Berlin (1985).
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.
27. W. Ziemer: Weakly Differentiable Functions, Springer-Verlag, Berlin (1989).