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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
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2299-3274
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Compactness of Special Functions of Bounded Higher Variation

Luigi Ambrosio / Francesco Ghiraldin
Published Online: 2013-01-04 | DOI: https://doi.org/10.2478/agms-2012-0001

Abstract

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].

Keywords: Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity

MSC: 49Q20; 49J45; 49Q15

  • R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. Google Scholar

  • 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. Google Scholar

  • G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J., 54(5):1411–1472, 2005. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • L. Ambrosio. A new proof of the SBV compactness theorem. Calc. Var. Partial Differential Equations, 3(1):127–137, 1995. 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. Google Scholar

  • 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. CrossrefGoogle Scholar

  • L. Ambrosio and B. Kirchheim. Currents in metric spaces. Acta Math., 185(1):1–80, 2000. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • L. Ambrosio and V. M. Tortorelli. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992. Google Scholar

  • J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77. Google Scholar

  • 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. Google Scholar

  • H. Brezis, J.-M. Coron, and E. H. Lieb. Harmonic maps with defects. Comm. Math. Phys., 107(4):649–705, 1986. Google Scholar

  • H. Brezis and H.-M. Nguyen. The Jacobian determinant revisited. Invent. Math., 185(1):17–54, 2011. Google Scholar

  • H. Brezis and L. Nirenberg. Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.), 1(2):197–263, 1995. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • G. Dal Maso. Generalised functions of bounded deformation. Preprint, 2011. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • C. De Lellis. Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A, 132(4):815–842, 2002. Google Scholar

  • C. De Lellis. Some remarks on the distributional Jacobian. Nonlinear Anal., 53(7-8):1101–1114, 2003. Google Scholar

  • 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. Google Scholar

  • T. De Pauw and R. Hardt. Rectifiable and flat G chains in a metric space. Amer. J. Math., 134(1):1–69, 2012. Google Scholar

  • G. De Philippis. Weak notions of Jacobian determinant and relaxation. ESAIM Control Optim. Calc. Var., 18(1):181– 207, 2012. Google Scholar

  • 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. Google Scholar

  • E. DiBenedetto. C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., 7(8):827– 850, 1983. Google Scholar

  • H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer- Verlag New York Inc., New York, 1969. Google Scholar

  • H. Federer. Flat chains with positive densities. Indiana Univ. Math. J., 35(2):413–424, 1986. Google Scholar

  • H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960. Google Scholar

  • W. H. Fleming. Flat chains over a finite coefficient group. Trans. Amer. Math. Soc., 121:160–186, 1966. Google Scholar

  • I. Fonseca, G. Leoni, and J. Malý. Weak continuity and lower semicontinuity results for determinants. Arch. Ration. Mech. Anal., 178(3):411–448, 2005. Google Scholar

  • N. Fusco and J. E. Hutchinson. A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math., 87(1):35–50, 1995. Google Scholar

  • F. Ghiraldin. Forthcoming. Google Scholar

  • 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. Google Scholar

  • R. Hardt, F. Lin, and C. Wang. The p-energy minimality of x/|x|. Comm. Anal. Geom., 6(1):141–152, 1998. CrossrefGoogle Scholar

  • R. Hardt and T. Rivière. Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(2):287–344, 2003. Google Scholar

  • A. D. Ioffe. On lower semicontinuity of integral functionals. I. SIAM J. Control Optimization, 15(4):521–538, 1977. CrossrefGoogle Scholar

  • A. D. Ioffe. On lower semicontinuity of integral functionals. II. SIAM J. Control Optimization, 15(6):991–1000, 1977. CrossrefGoogle Scholar

  • R. L. Jerrard and H. M. Soner. Functions of bounded higher variation. Indiana Univ. Math. J., 51(3):645–677, 2002. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • D. Mucci. A variational problem involving the distributional determinant. Riv. Math. Univ. Parma (N.S.), 1(2):321– 345, 2010. Google Scholar

  • D. Mucci. Graphs of vector valued maps: decomposition of the boundary. 2011. Google Scholar

  • S. Müller. Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math., 311(1):13–17, 1990. Google Scholar

  • S. Müller. On the singular support of the distributional determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(6):657–696, 1993. Google Scholar

  • 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. Google Scholar

  • D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989. Google Scholar

  • 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. Google Scholar

  • 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. Google Scholar

  • V. Šverák. Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal., 100(2):105–127, 1988. CrossrefGoogle Scholar

  • K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. Acta Math., 138(3-4):219–240, 1977. Google Scholar

  • B. White. Rectifiability of flat chains. Ann. of Math. (2), 150(1):165–184, 1999. Google Scholar

About the article

Received: 2012-10-16

Accepted: 2012-12-07

Published Online: 2013-01-04


Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 1–30, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2012-0001.

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