Majorisation with applications to the calculus of variations

Details Majorisation with applications to the calculus of variations Majorisation with applications to the calculus of variations

2001-05-05

arxiv.org/abs/math/0105044v4

appeared as "Four applications of majorization to convexity in the calculus of variations", Linear Algebra and its Application

Mathematics

Functional Analysis

13 pages

Scientific paper

10.1016/j.laa.2008.04.023

This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of possible non-polyconvex but multiplicative quasiconvex isotropic functions. This class is not contained in a well known theorem of Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and objective function to be polyconvex. We show here that there is a new way to prove directly the quasiconvexity (in the multiplicative form). Relevance of Schur convexity for the description of rank one convex hulls is explained.

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