Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2013 | 1 | 1-30

Tytuł artykułu

Compactness of Special Functions of Bounded Higher Variation

Treść / Zawartość

Warianty tytułu

Języki publikacji



Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m−n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].


  • Scuola Normale Superiore di Pisa,
    Piazza dei Cavalieri 7, I-56126, Pisa, Italy
  • Scuola Normale Superiore di Pisa,
    Piazza dei Cavalieri 7, I-56126, Pisa, Italy


  • R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.
  • R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.
  • G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J., 54(5):1411–1472, 2005.
  • F. Almgren. Deformations and multiple-valued functions. In Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 29–130. Amer. Math. Soc., Providence, RI, 1986.
  • L. Ambrosio. A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B (7), 3(4):857–881, 1989.
  • L. Ambrosio. A new proof of the SBV compactness theorem. Calc. Var. Partial Differential Equations, 3(1):127–137, 1995.
  • 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 Poincare (C) Non Linear Analysis, doi:10.1016/j.anihpc.2012.06.002, 2012. [Crossref]
  • L. Ambrosio and B. Kirchheim. Currents in metric spaces. Acta Math., 185(1):1–80, 2000.
  • L. Ambrosio and P. Tilli. Topics on analysis in metric spaces, volume 25 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004.
  • L. Ambrosio and V. M. Tortorelli. Approximation of functionals depending on jumps by elliptic functionals via г-convergence. Comm. Pure Appl. Math., 43(8):999–1036, 1990.
  • L. Ambrosio and V. M. Tortorelli. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992.
  • J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77.
  • A. Bressan. Hyperbolic systems of conservation laws, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem.
  • H. Brezis, J.-M. Coron, and E. H. Lieb. Harmonic maps with defects. Comm. Math. Phys., 107(4):649–705, 1986. [Crossref]
  • H. Brezis and H.-M. Nguyen. The Jacobian determinant revisited. Invent. Math., 185(1):17–54, 2011.
  • H. Brezis and L. Nirenberg. Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.), 1(2):197–263, 1995.
  • P. Celada and G. Dal Maso. Further remarks on the lower semicontinuity of polyconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire, 11(6):661–691, 1994.
  • R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes. Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9), 72(3):247–286, 1993.
  • C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010.
  • G. Dal Maso. Generalised functions of bounded deformation. Preprint, 2011.
  • E. De Giorgi and L. Ambrosio. New functionals in calculus of variations. In Nonsmooth optimization and related topics (Erice, 1988), volume 43 of Ettore Majorana Internat. Sci. Ser. Phys. Sci., pages 49–59. Plenum, New York, 1989.
  • E. De Giorgi, M. Carriero, and A. Leaci. Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal., 108(3):195–218, 1989.
  • C. De Lellis. Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A, 132(4):815–842, 2002.
  • C. De Lellis. Some remarks on the distributional Jacobian. Nonlinear Anal., 53(7-8):1101–1114, 2003.
  • C. De Lellis and F. Ghiraldin. An extension of the identity Det = det. C. R. Math. Acad. Sci. Paris, 348(17-18):973– 976, 2010.
  • T. De Pauw and R. Hardt. Rectifiable and flat G chains in a metric space. Amer. J. Math., 134(1):1–69, 2012.
  • G. De Philippis. Weak notions of Jacobian determinant and relaxation. ESAIM Control Optim. Calc. Var., 18(1):181– 207, 2012.
  • G. de Rham. Variétés différentiables. Formes, courants, formes harmoniques. Actualités Sci. Ind., no. 1222 = Publ. Inst. Math. Univ. Nancago III. Hermann et Cie, Paris, 1955.
  • E. DiBenedetto. C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., 7(8):827– 850, 1983.
  • H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer- Verlag New York Inc., New York, 1969.
  • H. Federer. Flat chains with positive densities. Indiana Univ. Math. J., 35(2):413–424, 1986.
  • H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960.
  • W. H. Fleming. Flat chains over a finite coefficient group. Trans. Amer. Math. Soc., 121:160–186, 1966.
  • I. Fonseca, G. Leoni, and J. Malý. Weak continuity and lower semicontinuity results for determinants. Arch. Ration. Mech. Anal., 178(3):411–448, 2005.
  • N. Fusco and J. E. Hutchinson. A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math., 87(1):35–50, 1995.
  • F. Ghiraldin. Forthcoming.
  • M. Giaquinta, G. Modica, and J. Soucek. Cartesian currents in the calculus of variations. I, II, volume 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.
  • R. Hardt, F. Lin, and C. Wang. The p-energy minimality of x/|x|. Comm. Anal. Geom., 6(1):141–152, 1998.
  • R. Hardt and T. Rivière. Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(2):287–344, 2003.
  • A. D. Ioffe. On lower semicontinuity of integral functionals. I. SIAM J. Control Optimization, 15(4):521–538, 1977.
  • A. D. Ioffe. On lower semicontinuity of integral functionals. II. SIAM J. Control Optimization, 15(6):991–1000, 1977.
  • R. L. Jerrard and H. M. Soner. Functions of bounded higher variation. Indiana Univ. Math. J., 51(3):645–677, 2002.
  • P. Marcellini. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3(5):391–409, 1986.
  • L. Modica and S. Mortola. Il limite nella 􀀀-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5), 14(3):526–529, 1977.
  • C. B. Morrey, Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York, 1966.
  • D. Mucci. A variational problem involving the distributional determinant. Riv. Math. Univ. Parma (N.S.), 1(2):321– 345, 2010.
  • D. Mucci. Graphs of vector valued maps: decomposition of the boundary. 2011.
  • S. Müller. Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math., 311(1):13–17, 1990.
  • S. Müller. On the singular support of the distributional determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(6):657–696, 1993.
  • S. Müller and S. J. Spector. An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal., 131(1):1–66, 1995.
  • D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989.
  • L. Schwartz. Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris, 1966.
  • E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
  • V. Šverák. Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal., 100(2):105–127, 1988.
  • K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. Acta Math., 138(3-4):219–240, 1977.
  • B. White. Rectifiability of flat chains. Ann. of Math. (2), 150(1):165–184, 1999.

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.