Compactness of Special Functions of Bounded Higher Variation

Luigi Ambrosio; Francesco Ghiraldin

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 1-30
  • ISSN: 2299-3274

Abstract

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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].

How to cite

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Luigi Ambrosio, and Francesco Ghiraldin. "Compactness of Special Functions of Bounded Higher Variation." Analysis and Geometry in Metric Spaces 1 (2013): 1-30. <http://eudml.org/doc/266762>.

@article{LuigiAmbrosio2013,
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].},
author = {Luigi Ambrosio, Francesco Ghiraldin},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity; higher codimension singularities; distributional Jacobian; Mumford-Shah type funtional},
language = {eng},
pages = {1-30},
title = {Compactness of Special Functions of Bounded Higher Variation},
url = {http://eudml.org/doc/266762},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Luigi Ambrosio
AU - Francesco Ghiraldin
TI - Compactness of Special Functions of Bounded Higher Variation
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 1
EP - 30
AB - 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].
LA - eng
KW - Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity; higher codimension singularities; distributional Jacobian; Mumford-Shah type funtional
UR - http://eudml.org/doc/266762
ER -

References

top
  1. R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. Zbl0314.46030
  2. 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. Zbl1098.46001
  3. G. Alberti, S. Baldo, and G. Orlandi. Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J., 54(5):1411–1472, 2005. Zbl1160.35013
  4. 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. 
  5. 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. Zbl0767.49001
  6. L. Ambrosio. A new proof of the SBV compactness theorem. Calc. Var. Partial Differential Equations, 3(1):127–137, 1995. Zbl0837.49011
  7. 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. Zbl0957.49001
  8. 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] 
  9. L. Ambrosio and B. Kirchheim. Currents in metric spaces. Acta Math., 185(1):1–80, 2000. Zbl0984.49025
  10. 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. Zbl1080.28001
  11. 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. Zbl0722.49020
  12. L. Ambrosio and V. M. Tortorelli. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992. Zbl0776.49029
  13. J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77. 
  14. 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. 
  15. H. Brezis, J.-M. Coron, and E. H. Lieb. Harmonic maps with defects. Comm. Math. Phys., 107(4):649–705, 1986. [Crossref] Zbl0608.58016
  16. H. Brezis and H.-M. Nguyen. The Jacobian determinant revisited. Invent. Math., 185(1):17–54, 2011. Zbl1230.46029
  17. H. Brezis and L. Nirenberg. Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.), 1(2):197–263, 1995. Zbl0852.58010
  18. 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. Zbl0833.49013
  19. 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. Zbl0864.42009
  20. 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. Zbl1196.35001
  21. G. Dal Maso. Generalised functions of bounded deformation. Preprint, 2011. Zbl1271.49029
  22. 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. Zbl0735.49001
  23. 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. Zbl0682.49002
  24. C. De Lellis. Some fine properties of currents and applications to distributional Jacobians. Proc. Roy. Soc. Edinburgh Sect. A, 132(4):815–842, 2002. Zbl1025.49029
  25. C. De Lellis. Some remarks on the distributional Jacobian. Nonlinear Anal., 53(7-8):1101–1114, 2003. Zbl1025.49030
  26. 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. Zbl1201.35088
  27. T. De Pauw and R. Hardt. Rectifiable and flat G chains in a metric space. Amer. J. Math., 134(1):1–69, 2012. Zbl1252.49070
  28. G. De Philippis. Weak notions of Jacobian determinant and relaxation. ESAIM Control Optim. Calc. Var., 18(1):181– 207, 2012. Zbl1242.49025
  29. 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. 
  30. E. DiBenedetto. C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., 7(8):827– 850, 1983. Zbl0539.35027
  31. H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer- Verlag New York Inc., New York, 1969. 
  32. H. Federer. Flat chains with positive densities. Indiana Univ. Math. J., 35(2):413–424, 1986. Zbl0611.49029
  33. H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960. Zbl0187.31301
  34. W. H. Fleming. Flat chains over a finite coefficient group. Trans. Amer. Math. Soc., 121:160–186, 1966. Zbl0136.03602
  35. I. Fonseca, G. Leoni, and J. Malý. Weak continuity and lower semicontinuity results for determinants. Arch. Ration. Mech. Anal., 178(3):411–448, 2005. Zbl1081.49013
  36. N. Fusco and J. E. Hutchinson. A direct proof for lower semicontinuity of polyconvex functionals. Manuscripta Math., 87(1):35–50, 1995. Zbl0874.49015
  37. F. Ghiraldin. Forthcoming. 
  38. 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. Zbl0914.49001
  39. R. Hardt, F. Lin, and C. Wang. The p-energy minimality of x/|x|. Comm. Anal. Geom., 6(1):141–152, 1998. Zbl0922.58015
  40. R. Hardt and T. Rivière. Connecting topological Hopf singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(2):287–344, 2003. Zbl1150.58004
  41. A. D. Ioffe. On lower semicontinuity of integral functionals. I. SIAM J. Control Optimization, 15(4):521–538, 1977. Zbl0361.46037
  42. A. D. Ioffe. On lower semicontinuity of integral functionals. II. SIAM J. Control Optimization, 15(6):991–1000, 1977. Zbl0379.46022
  43. R. L. Jerrard and H. M. Soner. Functions of bounded higher variation. Indiana Univ. Math. J., 51(3):645–677, 2002. Zbl1057.49036
  44. 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. Zbl0609.49009
  45. 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. Zbl0364.49006
  46. 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. 
  47. D. Mucci. A variational problem involving the distributional determinant. Riv. Math. Univ. Parma (N.S.), 1(2):321– 345, 2010. Zbl1222.49056
  48. D. Mucci. Graphs of vector valued maps: decomposition of the boundary. 2011. 
  49. S. Müller. Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math., 311(1):13–17, 1990. Zbl0717.46033
  50. S. Müller. On the singular support of the distributional determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(6):657–696, 1993. Zbl0792.46027
  51. 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. Zbl0836.73025
  52. D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989. Zbl0691.49036
  53. 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. 
  54. 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. Zbl0821.42001
  55. V. Šverák. Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal., 100(2):105–127, 1988. Zbl0659.73038
  56. K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. Acta Math., 138(3-4):219–240, 1977. 
  57. B. White. Rectifiability of flat chains. Ann. of Math. (2), 150(1):165–184, 1999. Zbl0965.49024

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