# Weak notions of Jacobian determinant and relaxation

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

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

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topDe 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- G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS)5 (2003) 275–311.
- 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).
- J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1976) 337–403.
- 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.
- F. Bethuel, The approximation problem for Sobolev maps between two manifolds. Acta Math.167 (1991) 153–206.
- 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.
- H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Mathematica (N.S.)1 (1995) 197–263.
- H. Brezis and L. Nirenberg, Degree theory and BMO. II. Compact manifolds with boundaries. Selecta Mathematica (N.S.)2 (1996) 309–368.
- H. Brezis, J.-M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys.107 (1986) 649–705.
- R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9)72 (1993) 247–286.
- 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.
- B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer, New York, second edition (2008).
- 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.
- C. De Lellis, Some fine properties of currents and applications to distributional Jacobians. Proc. R. Soc. Edinb. Sect. A132 (2002) 815–842.
- C. De Lellis, Some remarks on the distributional Jacobian. Nonlinear Anal.53 (2003) 1101–1114.
- C. De Lellis and F. Ghiraldin, An extension of Müller’s identity Det = det. C. R. Math. Acad. Sci. Paris348 (2010) 973–976.
- 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, Band153. 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).
- I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire14 (1997) 309–338.
- I. Fonseca and P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal.7 (1997) 57–81.
- I. Fonseca, N. Fusco and P. Marcellini, On the total variation of the Jacobian. J. Funct. Anal.207 (2004) 1–32.
- 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.
- 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.
- M. Giaquinta, G. Modica and J. Souček, Remarks on the degree theory. J. Funct. Anal.125 (1994) 172–200.
- 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).
- R.L. Jerrard and H.M. Soner, Functions of bounded higher variation. Indiana Univ. Math. J.51 (2002) 645–677.
- J. Malý, Lp-approximation of Jacobians. Comment. Math. Univ. Carolin.32 (1991) 659–666.
- P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math.51 (1985) 1–28.
- P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire3 (1986) 391–409.
- 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.
- D. Mucci, Remarks on the total variation of the Jacobian. NoDEA Nonlinear Differential Equations Appl.13 (2006) 223–233.
- D. Mucci, A variational problem involving the distributional determinant. Riv. Mat. Univ. Parma (to appear).
- S. Müller, Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math.412 (1990) 20–34.
- S. Müller, Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math.311 (1990) 13–17.
- S. Müller, On the singular support of the distributional determinant. Ann. Inst. Henri Poincaré Anal. Non Linéaire10 (1993) 657–696.
- S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal.131 (1995) 1–66.
- 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.
- S. Müller, S.J. Spector and Q. Tang, Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal.27 (1996) 959–976.
- E. Paolini, On the relaxed total variation of singular maps. Manuscr. Math.111 (2003) 499–512.
- A.C. Ponce and J. Van Schaftingen, Closure of smooth maps in W1, p(B3;S2). Differential Integral Equations22 (2009) 881–900.
- T. Schmidt, Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth. Arch. Rational Mech. Anal.193 (2009) 311–337.
- B. White, Existence of least-area mappings of N-dimensional domains. Ann. Math. (2)118 (1983) 179–185.

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