Skip to main content Accessibility help
×
Home

Variational approximation of a functional of Mumford–Shah type in codimension higher than one

  • Francesco Ghiraldin (a1)

Abstract

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].

Copyright

References

Hide All
[1] Acerbi, E. and Dal Maso, G., New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Equ. 2 (1994) 329371.
[2] R.A. Adams and J.J.F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edition. Elsevier/Academic Press, Amsterdam (2003).
[3] Alberti, G., Baldo, S. and Orlandi, G., Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5 (2003) 275311.
[4] Alberti, G., Baldo, S. and Orlandi, G., Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J. 54 (2005) 14111472.
[5] F. Almgren. 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.
[6] Ambrosio, L., Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990) 291322.
[7] Ambrosio, L., Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990) 439478.
[8] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000).
[9] L. Ambrosio and F. Ghiraldin, Flat chains of finite size in metric spaces. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis (2012).
[10] Ambrosio, L. and Ghiraldin, F., Compactness of special functions of bounded higher variation. Analysis and Geometry in Metric Spaces 1 (2013) 130.
[11] Ambrosio, L. and Kirchheim, B., Currents in metric spaces. Acta Math. 185 (2000) 180.
[12] Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 9991036.
[13] Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105123.
[14] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/1977) 337403.
[15] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices, vol. 13 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston, MA (1994).
[16] A. Braides, Γ -convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002).
[17] G. Dal Maso, An introduction to Γ-convergence. Progr. Nonlinear Differ. Eq. Appl., vol. 8. Birkhäuser Boston Inc., Boston, MA (1993).
[18] G. David, Singular sets of minimizers for the Mumford-Shah functional, vol. 233 of Progress in Mathematics. Birkhäuser Verlag, Basel (2005).
[19] De Giorgi, E., Carriero, M. and Leaci, A., Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195218.
[20] De Lellis, C., Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 815842.
[21] C. De Lellis and F. Ghiraldin, An extension of the identity Det = det. C. R. Acad. Sci. Paris Sér. I Math. (2010).
[22] De Pauw, T. and Hardt, R.. Rectifiable and flat G chains in a metric space. Amer. J. Math. 134 (2012) 169.
[23] Desenzani, N. and Fragalà, I., Concentration of Ginzburg-Landau energies with supercritical growth. SIAM J. Math. Anal. 38 (2006) 385413 (electronic).
[24] H. Federer, Geometric measure theory, vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Band. Springer-Verlag New York Inc., New York (1969).
[25] Federer, H., Flat chains with positive densities. Indiana Univ. Math. J. 35 (1986) 413424.
[26] Fleming, W. H., Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121 (1966) 160186.
[27] Fusco, N. and Hutchinson, J.E., A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math. 87 (1995) 3550.
[28] M. Giaquinta, G. Modica and J. Souček, 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).
[29] Enrico Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co. Inc., River Edge, NJ (2003). MR 1962933 (2004g:49003)
[30] Hardt, R. and Rivière, T., Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 287344.
[31] Jerrard, R.L. and Soner, H.M., Functions of bounded higher variation. Indiana Univ. Math. J. 51 (2002) 645677.
[32] E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14 of Amer. Math. Soc. Providence, RI, 2nd edition (2001).
[33] Modica, L. and Mortola, S., Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14 (1977) 526529.
[34] Modica, L. and Mortola, S., Un esempio di Γ -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285299.
[35] Morgan, F., Size-minimizing rectifiable currents. Invent. Math. 96 (1989) 333348.
[36] Müller, S., Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 1317.
[37] Müller, S. and Spector, S.J., An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995) 166.
[38] Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577685.
[39] E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, vol. 70 of Progress Non. Differ. Eqs. Appl. Birkhäuser Boston Inc., Boston, MA (2007).
[40] Šverák, V., Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105127.
[41] Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353372.
[42] White, B., Rectifiability of flat chains. Ann. Math. 150 (1999) 165184.
[43] W.P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989).

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed