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The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent

Published online by Cambridge University Press:  14 November 2011

Guy Bouchitté ,
Irene Fonseca  and
Jan Malý
Guy Bouchitté
Affiliation:
Département de Mathématiques, Université de Toulon et du Var—BP 132, 83957 La Garde Cedex, France
Irene Fonseca
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Jan Malý
Affiliation:
Faculty of Mathematics and Physics—KMA, Charles University, Sokolovská 83, 18600 Praha 8, Czech Republic
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Abstract

The characterisation of the bulk energy density of the relaxation in W1, P(Ω; ℝd) of a functional

is obtained for p > q – q/N, where u ∈ W1, P(Ω; ℝd), and f is a continuous function on the set of d × N matrices verifying

for some constant C > 0 and 1 ≦ q < + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 463 - 479
Copyright
Copyright © Royal Society of Edinburgh 1998

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