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On the Representation of Effective Energy Densities

Published online by Cambridge University Press:  15 August 2002

Christopher J. Larsen
Christopher J. Larsen*
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.; [email protected].
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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.

Keywords

Relaxationquasiconvexityintegral representation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 529 - 538
Copyright
© EDP Sciences, SMAI, 2000

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References

Choksi, R. and Fonseca, I., Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal. 138 (1997) 37-103. CrossRef
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989).
De Giorgi, E. and Ambrosio, L., 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.
Del Piero, G. and Owen, D.R., Structured deformations of continua. Arch. Rational Mech. Anal. 124 (1993) 99-155. CrossRef
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. CrossRef
Kristensen, J., Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653-710. CrossRef
Larsen, C.J., Quasiconvexification in W 1,1 and optimal jump microstructure in BV relaxation. SIAM J. Math. Anal. 29 (1998) 823-848. CrossRef
Müller, S., On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41 (1992) 295-301. CrossRef