The \(\Gamma\)-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension. (English) Zbl 1287.49013
Math. Models Methods Appl. Sci. 24, No. 6, 1091-1113 (2014); erratum ibid. 34, No. 11, 4987-4987 (2014).
Summary: In this paper, we generalizea one-dimensional equicoerciveness and \(\Gamma\)-convergence result for a second derivative perturbation of Perona-Malik type functionals to arbitrary dimensions. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a “cohesive” energy, that is, whose cost depends on the actual opening of the discontinuity.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
References:
| [1] | Alberti, G.Müller, S., Commun. Pure Appl. Math.54, 761 (2001), DOI: 10.1002/cpa.1013. · Zbl 1021.49012 |
| [2] | Alicandro, R.Braides, A.Gelli, M. S., Proc. Roy. Soc. Edinburgh Sec. A128, 1115 (1998), DOI: 10.1017/S0308210500027256. |
| [3] | Alicandro, R.Braides, A.Shah, J., Interfaces Free Bound1, 17 (1999). |
| [4] | Alicandro, R.Focardi, M., Commun. Contemp. Math.4, 685 (2002), DOI: 10.1142/S0219199702000816. |
| [5] | Alicandro, R.Gelli, M. S., Proc. Roy. Soc. Edinburgh Sec. A130, 449 (2000). · Zbl 0978.49014 |
| [6] | Ambrosio, L.; Dancer, N.; Buttazzo, G.; Marino, A.; Murthy, M. K. V., Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory, 2000, Springer |
| [7] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, 2000, Clarendon Press · Zbl 0957.49001 |
| [8] | Ambrosio, L.Lemenant, A.Royer-Carfagni, G., J. Elasticity110, 201 (2013), DOI: 10.1007/s10659-012-9390-5. |
| [9] | Ambrosio, L.Tortorelli, V. M., Boll. Unione Mat. Ital. VII Ser. B6, 105 (1992). · Zbl 0776.49029 |
| [10] | Bellettini, G.Fusco, G., Trans. Amer. Math. Soc.360, 4929 (2008), DOI: 10.1090/S0002-9947-08-04495-4. |
| [11] | Bellettini, G.Novaga, M.Paolini, E., SIAM J. Math. Anal.37, 1657 (2006), DOI: 10.1137/050625333. · Zbl 1109.35050 |
| [12] | Bellettini, G.et al., J. Differential Equations245, 892 (2008), DOI: 10.1016/j.jde.2008.05.003. |
| [13] | Bellettini, G.et al., Calcolo46, 221 (2009), DOI: 10.1007/s10092-009-0006-9. |
| [14] | Bouchitté, G.Dubs, C.Seppecher, P., C. R. Acad. Sci. Paris Sér. I Math.323, 1103 (1996). · Zbl 0859.49014 |
| [15] | Bouchitté, G.Dubs, C.Seppecher, P., Math. Models Methods Appl. Sci.10, 1073 (2000). · Zbl 1009.49012 |
| [16] | Bourdin, B.Francfort, G.Marigo, J.-J., J. Elasticity91, 5 (2008), DOI: 10.1007/s10659-007-9107-3. |
| [17] | Braides, A., Approximation of Free-Discontinuity Problems, 1694, 1998, Springer-Verlag · Zbl 0909.49001 |
| [18] | Braides, A., Γ-Convergence for Beginners, 22, 2002, Oxford Univ. Press · Zbl 1198.49001 |
| [19] | Chambolle, A., J. Math. Pures Appl.83, 929 (2004), DOI: 10.1016/j.matpur.2004.02.004. |
| [20] | A. Chambolle, Addendum to “An approximation result for special functions with bounded deformation” J. Math. Pures Appl. (9) 83 (2004) 929-954]: The N-dimensional case, J. Math. Pures Appl. 84 (2005) 137-145. · Zbl 1084.49038 |
| [21] | Chambolle, A.Giacomini, A.Ponsiglione, M., J. Funct. Anal.244, 134 (2007), DOI: 10.1016/j.jfa.2006.11.006. · Zbl 1110.74046 |
| [22] | Colombo, M.Gobbino, M., SIAM J. Math. Anal.43, 2564 (2011), DOI: 10.1137/100818698. |
| [23] | Conti, S.Fonseca, I.Leoni, G., Commun. Pure Appl. Math.55, 857 (2002), DOI: 10.1002/cpa.10035. |
| [24] | Conti, S.Schweizer, B., Commun. Pure Appl. Math.59, 830 (2006), DOI: 10.1002/cpa.20115. |
| [25] | Cortesani, G., Arch. Rational Mech. Anal.144, 357 (1998), DOI: 10.1007/s002050050121. |
| [26] | Cortesani, G.Toader, R., Nonlinear Anal. Real World Appl.38, 585 (1999), DOI: 10.1016/S0362-546X(98)00132-1. |
| [27] | Dal Maso, G., An Introduction to Γ-Convergence, 1993, Birkhäuser · Zbl 0816.49001 |
| [28] | Dal Maso, G.Iurlano, F., Commun. Pure Appl. Anal.12, 1657 (2013), DOI: 10.3934/cpaa.2013.12.1657. · Zbl 1345.74097 |
| [29] | De Giorgi, E., Duke Math. J.81, 255 (1995), DOI: 10.1215/S0012-7094-96-08114-4. |
| [30] | Esedoglu, S., Commun. Pure Appl. Math.54, 1442 (2001). · Zbl 1031.68133 |
| [31] | Evans, L. C.; Gariepy, G., Measure Theory and Fine Properties of Functions, 1992, CRC Press · Zbl 0804.28001 |
| [32] | Fonseca, I.Mantegazza, C., SIAM J. Math. Anal.31, 1121 (2000), DOI: 10.1137/S0036141099356830. · Zbl 0958.49007 |
| [33] | Francfort, G. A.Marigo, J. J., J. Mech. Phys. Solids46, 1319 (1998), DOI: 10.1016/S0022-5096(98)00034-9. |
| [34] | Ghisi, M.Gobbino, M., Math. Ann.337, 557 (2007), DOI: 10.1007/s00208-006-0047-1. |
| [35] | Giusti, E., Minimal Surfaces and Functions of Bounded Variation, 80, 1984, Birkhäuser · Zbl 0545.49018 |
| [36] | Gobbino, M., Commun. Pure Appl. Math.51, 197 (1998). · Zbl 0888.49013 |
| [37] | Gobbino, M., Comm. Partial Differential Equations32, 719 (2007), DOI: 10.1080/03605300600634981. |
| [38] | Gobbino, M.Mora, M. G., Proc. Roy. Soc. Edinburgh Sec. A131, 567 (2001), DOI: 10.1017/S0308210500001001. |
| [39] | Kawohl, B.Kutev, N., Math. Ann.311, 107 (1998), DOI: 10.1007/s002080050179. |
| [40] | Kichenassamy, S., SIAM J. Appl. Math.57, 1328 (2001), DOI: 10.1137/S003613999529558X. |
| [41] | Morini, M., SIAM J. Math. Anal.35, 759 (2003), DOI: 10.1137/S0036141001395388. |
| [42] | Mumford, D.Shah, J., Commun. Pure Appl. Math.42, 577 (1989), DOI: 10.1002/cpa.3160420503. |
| [43] | Oudet, E.Santambrogio, F., Arch. Rational Mech. Anal.201, 115 (2011), DOI: 10.1007/s00205-011-0402-6. |
| [44] | Perona, P.Malik, J., IEEE Trans. Pattern Anal. Mach. Intell.12, 629 (1990), DOI: 10.1109/34.56205. |
| [45] | Slemrod, M., J. Dynam. Differential Equations3, 1 (1991), DOI: 10.1007/BF01049487. |
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