In this paper we consider a new kind of Mumford-Shah functional E(u, Ω) for maps u : ℝm → ℝn with m ≥ n. The most important novelty is that the energy features a singular set Su of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy E(u, Ω) via Γ -convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math. 43 (1990) 999-1036].
Mots clés : jacobian, Γ-convergence, higher codimension, Mumford-Shah, Ginzburg-Landau, phase transition
@article{COCV_2014__20_1_190_0,
author = {Ghiraldin, Francesco},
title = {Variational approximation of a functional of {Mumford-Shah} type in codimension higher than one},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {190--221},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013061},
mrnumber = {3182697},
zbl = {1286.49054},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013061/}
}
TY - JOUR AU - Ghiraldin, Francesco TI - Variational approximation of a functional of Mumford-Shah type in codimension higher than one JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 190 EP - 221 VL - 20 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013061/ DO - 10.1051/cocv/2013061 LA - en ID - COCV_2014__20_1_190_0 ER -
%0 Journal Article %A Ghiraldin, Francesco %T Variational approximation of a functional of Mumford-Shah type in codimension higher than one %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 190-221 %V 20 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013061/ %R 10.1051/cocv/2013061 %G en %F COCV_2014__20_1_190_0
Ghiraldin, Francesco. Variational approximation of a functional of Mumford-Shah type in codimension higher than one. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 190-221. doi : 10.1051/cocv/2013061. http://www.numdam.org/articles/10.1051/cocv/2013061/
[1] and , New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Equ. 2 (1994) 329-371. | MR | Zbl
[2] and , Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edition. Elsevier/Academic Press, Amsterdam (2003). | MR | Zbl
[3] , and , Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5 (2003) 275-311. | MR | Zbl
[4] , and , Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J. 54 (2005) 1411-1472. | MR | Zbl
[5] . Deformations and multiple-valued functions, in Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), vol. 44 of Proc. Sympos. Pure Math. Amer. Math. Soc. Providence, RI (1986) 29-130. | MR | Zbl
[6] , Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990) 291-322. | MR | Zbl
[7] , Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990) 439-478. | Numdam | MR | Zbl
[8] , and , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). | MR | Zbl
[9] and , Flat chains of finite size in metric spaces. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis (2012). | Numdam | MR | Zbl
[10] and , Compactness of special functions of bounded higher variation. Analysis and Geometry in Metric Spaces 1 (2013) 1-30. | MR | Zbl
[11] and , Currents in metric spaces. Acta Math. 185 (2000) 1-80. | MR | Zbl
[12] and , Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. | Zbl
[13] and , On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105-123. | MR | Zbl
[14] , Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/1977) 337-403. | MR | Zbl
[15] , and , Ginzburg-Landau vortices, vol. 13 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston, MA (1994). | MR | Zbl
[16] , Γ-convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002). | MR | Zbl
[17] , An introduction to Γ-convergence. Progr. Nonlinear Differ. Eq. Appl., vol. 8. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl
[18] , Singular sets of minimizers for the Mumford-Shah functional, vol. 233 of Progress in Mathematics. Birkhäuser Verlag, Basel (2005). | MR | Zbl
[19] , and , Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195-218. | MR | Zbl
[20] , Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 815-842. | MR | Zbl
[21] and , An extension of the identity Det = det. C. R. Acad. Sci. Paris Sér. I Math. (2010). | Zbl
[22] and . Rectifiable and flat G chains in a metric space. Amer. J. Math. 134 (2012) 1-69. | MR | Zbl
[23] and , Concentration of Ginzburg-Landau energies with supercritical growth. SIAM J. Math. Anal. 38 (2006) 385-413 (electronic). | MR | Zbl
[24] , Geometric measure theory, vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Band. Springer-Verlag New York Inc., New York (1969). | MR | Zbl
[25] , Flat chains with positive densities. Indiana Univ. Math. J. 35 (1986) 413-424. | MR | Zbl
[26] , Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121 (1966) 160-186. | MR | Zbl
[27] and , A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math. 87 (1995) 35-50. | MR | Zbl
[28] , and , Cartesian currents in the calculus of variations. I, II, vol. 37, 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin (1998). | MR | Zbl
[29] , Direct methods in the calculus of variations. World Scientific Publishing Co. Inc., River Edge, NJ (2003). MR 1962933 (2004g:49003) | MR | Zbl
[30] and , Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 287-344. | Numdam | MR | Zbl
[31] and , Functions of bounded higher variation. Indiana Univ. Math. J. 51 (2002) 645-677. | MR | Zbl
[32] and , Analysis, Graduate Studies in Mathematics, vol. 14 of Amer. Math. Soc. Providence, RI, 2nd edition (2001). | MR | Zbl
[33] and , Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14 (1977) 526-529. | MR | Zbl
[34] and , Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR | Zbl
[35] , Size-minimizing rectifiable currents. Invent. Math. 96 (1989) 333-348. | MR | Zbl
[36] , Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 13-17. | MR | Zbl
[37] and , An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995) 1-66. | MR | Zbl
[38] and , Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. | MR | Zbl
[39] and , Vortices in the magnetic Ginzburg-Landau model, vol. 70 of Progress Non. Differ. Eqs. Appl. Birkhäuser Boston Inc., Boston, MA (2007). | MR | Zbl
[40] , Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105-127. | MR | Zbl
[41] . Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353-372. | MR | Zbl
[42] , Rectifiability of flat chains. Ann. Math. 150 (1999) 165-184. | MR | Zbl
[43] , Weakly differentiable functions, Sobolev spaces and functions of bounded variation, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989). | MR | Zbl
Cité par Sources :
