Nonlocal approximation of nonisotropic free-discontinuity problems

SIAM Journal on Applied Mathematics

Volumn 59, Issue 4, 1999, Pages 1507-1519

  Cortesani, Guido ^a   Toader, Rodica ^b  

^a SISSA   (Italy)

^b UNIVERSITY OF LISBON   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; INTEGRAL EQUATIONS; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

FREE-DISCONTINUITY PROBLEMS; NONLOCAL INTEGRAL FUNCTIONALS; WULFF SET;

APPROXIMATION THEORY;

EID: 0032628318     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139997327691     Document Type: Article

References (34)

1. R. ALICANDRO, A. BRAIDES, AND M.S. GELLI, Free-Dis continuity Problems Generated by Singular Perturbation, Preprint SISSA n. 7/97/M, Trieste, 1997.
2. F.J. ALMGREN, JR., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., 4 (1976), no. 165.
3. L. AMBROSIO, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital., B, 3 (1989), pp. 857-881.
4. L. AMBROSIO, Variational problems in SBV and image segmentation, Acta Appl. Math., 17 (1989), pp. 1-40.
5. L. AMBROSIO, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), pp. 291-322.
6. k), Nonlinear Anal., 23 (1994), pp. 405-425.
7. L. AMBROSIO, The space SBV(Ω) and free discontinuity problems, in Variational and Free Boundary Problems, A. Friedman and J. Spruch, eds., Springer-Verlag, New York, 1994, pp. 29-45.
8. L. AMBROSIO AND A. BRAIDES, Functionals defined on partitions of sets of finite perimeter, I: Integral representation and Γ-convergence, J. Math. Pures Appl., 69 (1990), pp. 285-305.
9. L. AMBROSIO AND A. BRAIDES, Functionals defined on partitions of sets of finite perimeter, II: Semicontinuity, relaxation and homogenization, J. Math. Pures Appl., 69 (1990), pp. 307-333.
10. L. AMBROSIO AND V.M. TORTORELLI, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999-1036.
11. m), J. Math. Pures Appl., 75 (1996), pp. 595-626.
12. A. BRAIDES AND A. COSCIA, The interaction between bulk energy and surface energy in multiple integrals, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), pp. 737-756.
13. A. BRAIDES AND G. DAL MASO, Non-local approximation of the Mumford-Shah functional, Calc. Var., 5 (1997), pp. 293-322.
14. A. BRAIDES AND A. GARRONI, A Remark on the Non-Local Approximation of Free-Discontinuity Problems, Preprint SISSA n. 8/97/M, Trieste, 1997.
15. G. CARRIERO, A. LEACI, AND F. TOMARELLI, Plastic free discontinuities and special bounded Hessian, C. R. Acad. Sci. Paris Sér. I, 324 (1992), pp. 595-600.
16. G. CORTESANI, Sequences of Non-Local Functionals which Approximate Free-Discontinuity Problems, Preprint SISSA n. 177/96/M, Trieste, 1996.
17. G. DAL MASO, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.
18. C. DAVINI, A proposal for a continuum theory of defective crystals, Arch. Rational Mech. Anal., 96 (1986), pp. 295-317.
19. E. DE GIORGI AND L. AMBROSIO, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend., 82 (1988), pp. 199-210.
20. A. DINGHAS, Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Krist., 105 (1944), pp. 304-314.
21. L.C. EVANS AND R.F. GARIEPY, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
22. H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969.
23. I. FONSECA, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A, 432 (1991), pp. 125-145.
24. I. FONSECA AND S. MÜLLER, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), pp. 125-136.
25. I. FONSECA AND G. PARRY, Equilibrium configurations of defective crystals, Arch. Rational Mech. Anal., 120 (1992), pp. 245-283.
26. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel, 1993.
27. M. GOBBINO, Non-Local Approximation of the Mumford-Shah Functional, Preprint, Universitá di Parma, 1996.
28. C. HERRING, Some theorems on the free energies of crystal surfaces, Phys. Rev., 82 (1951), pp. 87-93.
29. J.M. MOREL AND S. SOLIMINI, Variational Models in Image Segmentation, Birkhäuser, Boston, 1995.
30. D. MUMFORD AND J. SHAH, Boundary detection by minimizing functionals, I, in Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco, 1985.
31. D. MUMFORD AND J. SHAH, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 17 (1989), pp. 577-685.
32. E.G. VIRGA, Drops of nematic liquid crystals, Arch. Rational Mech. Anal., 107 (1989), pp. 371-390.
33. G. WULFF, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen, Z. Krist., 34 (1901), pp. 449-530.
34. W.P. ZIEMER, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989.