Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T15:23:47.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak notions of Jacobian determinant and relaxation

Published online by Cambridge University Press:  02 December 2010

Guido De Philippis
Guido De Philippis*
Affiliation:
Scuola Normale Superiore, P.za dei Cavalieri 7, 56100 Pisa, Italy. guido.dephilippis@sns.it
Get access

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.

Keywords

Distributional determinanttopological degreerelaxation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 181 - 207
Copyright
© EDP Sciences, SMAI, 2010

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

Alberti, G., Baldo, S. and Orlandi, G., Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5 (2003) 275–311. 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).
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976) 337–403. Google Scholar
Bethuel, F., A characterization of maps in H 1(B 3, S 2) which can be approximated by smooth maps. Ann. Inst. Henri Poincaré Anal. Non Linéaire 7 (1990) 269–286. Google Scholar
Bethuel, F., The approximation problem for Sobolev maps between two manifolds. Acta Math. 167 (1991) 153–206. Google Scholar
Bouchitté, G., Fonseca, I. and Malý, J., The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. R. Soc. Edinb. Sect. A 128 (1998) 463–479. Google Scholar
Brezis, H. and Nirenberg, L., Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Mathematica (N.S.) 1 (1995) 197–263. Google Scholar
Brezis, H. and Nirenberg, L., Degree theory and BMO. II. Compact manifolds with boundaries. Selecta Mathematica (N.S.) 2 (1996) 309–368. Google Scholar
Brezis, H., Coron, J.-M. and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649–705. Google Scholar
Coifman, R., Lions, P.-L., Meyer, Y. and Semmes, S., Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993) 247–286. Google Scholar
Conti, S. and De Lellis, C., Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003) 521–549. Google Scholar
B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer, New York, second edition (2008).
Dacorogna, B. and Marcellini, P., Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 393–396. Google Scholar
De Lellis, C., Some fine properties of currents and applications to distributional Jacobians. Proc. R. Soc. Edinb. Sect. A 132 (2002) 815–842. Google Scholar
De Lellis, C., Some remarks on the distributional Jacobian. Nonlinear Anal. 53 (2003) 1101–1114. Google Scholar
De Lellis, C. and Ghiraldin, F., An extension of Müller’s identity Det = det. C. R. Math. Acad. Sci. Paris 348 (2010) 973–976. Google Scholar
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969).
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).
Fonseca, I. and Malý, J., Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309–338. Google Scholar
Fonseca, I. and Marcellini, P., Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal. 7 (1997) 57–81. Google Scholar
Fonseca, I., Fusco, N. and Marcellini, P., On the total variation of the Jacobian. J. Funct. Anal. 207 (2004) 1–32. Google Scholar
Fonseca, I., Fusco, N. and Marcellini, P., Topological degree, Jacobian determinants and relaxation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005) 187–250. Google Scholar
Giaquinta, M., Modica, G. and Souček, J., Graphs of finite mass which cannot be approximated in area by smooth graphs. Manuscr. Math. 78 (1993) 259–271. Google Scholar
Giaquinta, M., Modica, G. and Souček, J., Remarks on the degree theory. J. Funct. Anal. 125 (1994) 172–200. Google Scholar
M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations I, Cartesian currents. Springer-Verlag, Berlin (1998).
A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge (2002).
Jerrard, R.L. and Soner, H.M., Functions of bounded higher variation. Indiana Univ. Math. J. 51 (2002) 645–677. Google Scholar
Malý, J., L p-approximation of Jacobians. Comment. Math. Univ. Carolin. 32 (1991) 659–666. Google Scholar
Marcellini, P., Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math. 51 (1985) 1–28. Google Scholar
Marcellini, P., On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 3 (1986) 391–409. Google Scholar
P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, in Partial differential equations and the calculus of variations II, Progr. Nonlinear Differential Equations Appl. 2, Birkhäuser Boston, Boston, MA (1989) 767–786.
Mucci, D., Remarks on the total variation of the Jacobian. NoDEA Nonlinear Differential Equations Appl. 13 (2006) 223–233. Google Scholar
D. Mucci, A variational problem involving the distributional determinant. Riv. Mat. Univ. Parma (to appear).
Müller, S., Higher integrability of determinants and weak convergence in L 1. J. Reine Angew. Math. 412 (1990) 20–34. 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., On the singular support of the distributional determinant. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 (1993) 657–696. 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
Müller, S., Tang, Q. and Yan, B.S., On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 11 (1994) 217–243. Google Scholar
Müller, S., Spector, S.J. and Tang, Q., Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal. 27 (1996) 959–976. Google Scholar
Paolini, E., On the relaxed total variation of singular maps. Manuscr. Math. 111 (2003) 499–512. Google Scholar
Ponce, A.C. and Van Schaftingen, J., Closure of smooth maps in W 1, p(B 3;S 2). Differential Integral Equations 22 (2009) 881–900. Google Scholar
Schmidt, T., Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth. Arch. Rational Mech. Anal. 193 (2009) 311–337. Google Scholar
White, B., Existence of least-area mappings of N-dimensional domains. Ann. Math. (2) 118 (1983) 179–185. Google Scholar