Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T18:12:10.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Relaxation for some dynamical problems

Published online by Cambridge University Press:  14 November 2011

B. Dacorogna
B. Dacorogna
Affiliation:
Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Suisse
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this article, we study the functional Where Ω ⊂ ĝn is a bounded open set and u: Ω ×(0, T)→ ĝm and when F: Rnm →R fails to be quasiconvex. We show that with respect to strong convergence of ∂u/∂t and weak convergence of ∇×u, the above functional behaves as where QF is the lower quasiconvex envelope of F.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 39 - 52
Copyright
Copyright © Royal Society of Edinburgh 1985

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

1Dacorogna, B.. Quasiconvexity and relaxation of non convex problems in thecalculus of variations. J. Funct. Anal. 46 (1982), 102–118.CrossRefGoogle Scholar
2Dacorogna, B.. Weak continuity and weak lower semicontinuity of nonlinear functionals. Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).Google Scholar
3Perna, R. J. Di. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.Google Scholar
4Ekeland, I. and Témam, R.. Analyse convexe et problèmes variationnels (Paris: Dunod, Gauthier- Villars, 1976).Google Scholar
5Morrey, C. B.. Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
6Morrey, C. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).Google Scholar
7Shearer, M.. Admissibility criteria for shock wave solutions of a system of conservations laws of mixed type. Proc. Roy. Soc. Edinburgh Sect. A 93 (1983), 233–244.CrossRefGoogle Scholar