Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T11:24:20.569Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  27 January 2014

Francesco Ghiraldin
Francesco Ghiraldin*
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy. f.ghiraldin@sns.it
Get access

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].

Keywords

JacobianΓ-convergencehigher codimensionMumford–ShahGinzburg–Landauphase transition
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 190 - 221
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and Dal Maso, G., New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Equ. 2 (1994) 329–371. Google Scholar
R.A. Adams and J.J.F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edition. Elsevier/Academic Press, Amsterdam (2003).
Alberti, G., Baldo, S. and Orlandi, G., Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5 (2003) 275–311. Google Scholar
Alberti, G., Baldo, S. and Orlandi, G., Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J. 54 (2005) 1411–1472. Google Scholar
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.
Ambrosio, L., Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990) 291–322. Google Scholar
Ambrosio, L., Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990) 439–478. Google Scholar
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).
L. Ambrosio and F. Ghiraldin, Flat chains of finite size in metric spaces. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis (2012).
Ambrosio, L. and Ghiraldin, F., Compactness of special functions of bounded higher variation. Analysis and Geometry in Metric Spaces 1 (2013) 1–30. Google Scholar
Ambrosio, L. and Kirchheim, B., Currents in metric spaces. Acta Math. 185 (2000) 1–80. Google Scholar
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999–1036. Google Scholar
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105–123. Google Scholar
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/1977) 337–403. Google Scholar
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).
A. Braides, Γ -convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002).
G. Dal Maso, An introduction to Γ-convergence. Progr. Nonlinear Differ. Eq. Appl., vol. 8. Birkhäuser Boston Inc., Boston, MA (1993).
G. David, Singular sets of minimizers for the Mumford-Shah functional, vol. 233 of Progress in Mathematics. Birkhäuser Verlag, Basel (2005).
De Giorgi, E., Carriero, M. and Leaci, A., Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195–218. Google Scholar
De Lellis, C., Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 815–842. Google Scholar
C. De Lellis and F. Ghiraldin, An extension of the identity Det = det. C. R. Acad. Sci. Paris Sér. I Math. (2010).
De Pauw, T. and Hardt, R.. Rectifiable and flat G chains in a metric space. Amer. J. Math. 134 (2012) 1–69. Google Scholar
Desenzani, N. and Fragalà, I., Concentration of Ginzburg-Landau energies with supercritical growth. SIAM J. Math. Anal. 38 (2006) 385–413 (electronic). Google Scholar
H. Federer, Geometric measure theory, vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Band. Springer-Verlag New York Inc., New York (1969).
Federer, H., Flat chains with positive densities. Indiana Univ. Math. J. 35 (1986) 413–424. Google Scholar
Fleming, W. H., Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121 (1966) 160–186. Google Scholar
Fusco, N. and Hutchinson, J.E., A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math. 87 (1995) 35–50. Google Scholar
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).
Enrico Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co. Inc., River Edge, NJ (2003). MR 1962933 (2004g:49003)
Hardt, R. and Rivière, T., Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 287–344. Google Scholar
Jerrard, R.L. and Soner, H.M., Functions of bounded higher variation. Indiana Univ. Math. J. 51 (2002) 645–677. Google Scholar
E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14 of Amer. Math. Soc. Providence, RI, 2nd edition (2001).
Modica, L. and Mortola, S., Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14 (1977) 526–529. Google Scholar
Modica, L. and Mortola, S., Un esempio di Γ −-convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. Google Scholar
Morgan, F., Size-minimizing rectifiable currents. Invent. Math. 96 (1989) 333–348. Google Scholar
Müller, S., Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 13–17. Google Scholar
Müller, S. and Spector, S.J., An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995) 1–66. Google Scholar
Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577–685. Google Scholar
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).
Šverák, V., Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105–127. Google Scholar
Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353–372. Google Scholar
White, B., Rectifiability of flat chains. Ann. Math. 150 (1999) 165–184. Google Scholar
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).