On the Representation of Effective Energy Densities

Christopher J. Larsen

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 5, page 529-538
  • ISSN: 1292-8119

Abstract

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We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.

How to cite

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Larsen, Christopher J.. "On the Representation of Effective Energy Densities." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 529-538. <http://eudml.org/doc/197301>.

@article{Larsen2010,
abstract = { We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation. },
author = {Larsen, Christopher J.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Relaxation; quasiconvexity; integral representation.; relaxation; integral representation; bulk and surface integrals},
language = {eng},
month = {3},
pages = {529-538},
publisher = {EDP Sciences},
title = {On the Representation of Effective Energy Densities},
url = {http://eudml.org/doc/197301},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Larsen, Christopher J.
TI - On the Representation of Effective Energy Densities
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 529
EP - 538
AB - We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.
LA - eng
KW - Relaxation; quasiconvexity; integral representation.; relaxation; integral representation; bulk and surface integrals
UR - http://eudml.org/doc/197301
ER -

References

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  1. R. Choksi and I. Fonseca, Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal.138 (1997) 37-103.  Zbl0891.73078
  2. B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989).  Zbl0703.49001
  3. E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni. Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Suppl.82 (1988) 199-210.  
  4. G. Del Piero and D.R. Owen, Structured deformations of continua. Arch. Rational Mech. Anal.124 (1993) 99-155.  Zbl0795.73005
  5. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).  Zbl0804.28001
  6. I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal.29 (1998) 736-756.  Zbl0920.49009
  7. J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann.313 (1999) 653-710.  Zbl0924.49012
  8. C.J. Larsen, Quasiconvexification in W1,1 and optimal jump microstructure in BV relaxation. SIAM J. Math. Anal.29 (1998) 823-848.  Zbl0915.49005
  9. S. Müller, On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J.41 (1992) 295-301.  Zbl0736.26006

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