Weak notions of Jacobian determinant and relaxation

Guido De Philippis

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 181-207
  • ISSN: 1292-8119

Abstract

top
In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.

How to cite

top

De Philippis, Guido. "Weak notions of Jacobian determinant and relaxation." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 181-207. <http://eudml.org/doc/221907>.

@article{DePhilippis2012,
abstract = {In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals. },
author = {De Philippis, Guido},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Distributional determinant; topological degree; relaxation; distributional determinant},
language = {eng},
month = {2},
number = {1},
pages = {181-207},
publisher = {EDP Sciences},
title = {Weak notions of Jacobian determinant and relaxation},
url = {http://eudml.org/doc/221907},
volume = {18},
year = {2012},
}

TY - JOUR
AU - De Philippis, Guido
TI - Weak notions of Jacobian determinant and relaxation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 181
EP - 207
AB - In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.
LA - eng
KW - Distributional determinant; topological degree; relaxation; distributional determinant
UR - http://eudml.org/doc/221907
ER -

References

top
  1. G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS)5 (2003) 275–311.  Zbl1033.46028
  2. 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
  3. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1976) 337–403.  Zbl0368.73040
  4. F. Bethuel, A characterization of maps in H1(B3,S2) which can be approximated by smooth maps. Ann. Inst. Henri Poincaré Anal. Non Linéaire7 (1990) 269–286.  
  5. F. Bethuel, The approximation problem for Sobolev maps between two manifolds. Acta Math.167 (1991) 153–206.  Zbl0756.46017
  6. G. Bouchitté, I. Fonseca and J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. R. Soc. Edinb. Sect. A128 (1998) 463–479.  Zbl0907.49008
  7. H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Mathematica (N.S.)1 (1995) 197–263.  Zbl0852.58010
  8. H. Brezis and L. Nirenberg, Degree theory and BMO. II. Compact manifolds with boundaries. Selecta Mathematica (N.S.)2 (1996) 309–368.  Zbl0868.58017
  9. H. Brezis, J.-M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys.107 (1986) 649–705.  Zbl0608.58016
  10. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9)72 (1993) 247–286.  Zbl0864.42009
  11. S. Conti and C. De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5)2 (2003) 521–549.  Zbl1114.74004
  12. B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer, New York, second edition (2008).  Zbl1140.49001
  13. B. Dacorogna and P. Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants. C. R. Acad. Sci. Paris Sér. I Math.311 (1990) 393–396.  Zbl0723.49007
  14. C. De Lellis, Some fine properties of currents and applications to distributional Jacobians. Proc. R. Soc. Edinb. Sect. A132 (2002) 815–842.  Zbl1025.49029
  15. C. De Lellis, Some remarks on the distributional Jacobian. Nonlinear Anal.53 (2003) 1101–1114.  Zbl1025.49030
  16. C. De Lellis and F. Ghiraldin, An extension of Müller’s identity Det = det. C. R. Math. Acad. Sci. Paris348 (2010) 973–976.  Zbl1201.35088
  17. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).  Zbl0804.28001
  18. H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band153. Springer-Verlag New York Inc., New York (1969).  
  19. I. Fonseca and W. Gangbo, Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1995).  Zbl0852.47030
  20. I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire14 (1997) 309–338.  Zbl0868.49011
  21. I. Fonseca and P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal.7 (1997) 57–81.  Zbl0915.49011
  22. I. Fonseca, N. Fusco and P. Marcellini, On the total variation of the Jacobian. J. Funct. Anal.207 (2004) 1–32.  Zbl1041.49016
  23. I. Fonseca, N. Fusco and P. Marcellini, Topological degree, Jacobian determinants and relaxation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)8 (2005) 187–250.  Zbl1177.49066
  24. M. Giaquinta, G. Modica and J. Souček, Graphs of finite mass which cannot be approximated in area by smooth graphs. Manuscr. Math.78 (1993) 259–271.  Zbl0796.58006
  25. M. Giaquinta, G. Modica and J. Souček, Remarks on the degree theory. J. Funct. Anal.125 (1994) 172–200.  Zbl0822.55003
  26. M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations I, Cartesian currents. Springer-Verlag, Berlin (1998).  Zbl0914.49001
  27. A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge (2002).  Zbl1044.55001
  28. R.L. Jerrard and H.M. Soner, Functions of bounded higher variation. Indiana Univ. Math. J.51 (2002) 645–677.  Zbl1057.49036
  29. J. Malý, Lp-approximation of Jacobians. Comment. Math. Univ. Carolin.32 (1991) 659–666.  Zbl0753.46024
  30. P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math.51 (1985) 1–28.  Zbl0573.49010
  31. P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire3 (1986) 391–409.  Zbl0609.49009
  32. P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, in Partial differential equations and the calculus of variationsII, Progr. Nonlinear Differential Equations Appl.2, Birkhäuser Boston, Boston, MA (1989) 767–786.  
  33. D. Mucci, Remarks on the total variation of the Jacobian. NoDEA Nonlinear Differential Equations Appl.13 (2006) 223–233.  Zbl1117.49017
  34. D. Mucci, A variational problem involving the distributional determinant. Riv. Mat. Univ. Parma (to appear).  Zbl1222.49056
  35. S. Müller, Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math.412 (1990) 20–34.  Zbl0713.49004
  36. S. Müller, Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math.311 (1990) 13–17.  
  37. S. Müller, On the singular support of the distributional determinant. Ann. Inst. Henri Poincaré Anal. Non Linéaire10 (1993) 657–696.  Zbl0792.46027
  38. S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal.131 (1995) 1–66.  Zbl0836.73025
  39. S. Müller, Q. Tang and B.S. Yan, On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré Anal. Non Linéaire11 (1994) 217–243.  Zbl0863.49002
  40. S. Müller, S.J. Spector and Q. Tang, Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal.27 (1996) 959–976.  Zbl0855.73028
  41. E. Paolini, On the relaxed total variation of singular maps. Manuscr. Math.111 (2003) 499–512.  Zbl1023.49010
  42. A.C. Ponce and J. Van Schaftingen, Closure of smooth maps in W1, p(B3;S2). Differential Integral Equations22 (2009) 881–900.  Zbl1240.46063
  43. T. Schmidt, Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth. Arch. Rational Mech. Anal.193 (2009) 311–337.  Zbl1173.49032
  44. B. White, Existence of least-area mappings of N-dimensional domains. Ann. Math. (2)118 (1983) 179–185.  Zbl0526.49029

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.