On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

• Published in: Calculus of variations and partial differential equations 2013-11, Vol.48 (3-4), p.625-665
• Main Author: Lorent, Andrew
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematical functions; Mathematics; Mathematics and Statistics; Partial differential equations; Systems Theory; Theorems; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-012-0566-4

We consider the following question: Given a connected open domain , suppose with det , det a.e. are such that a.e. , does this imply a global relation of the form a.e. in Ω where ? If u , v are C 1 it is an exercise to see this true, if we show this is false. In Theorem 1 we prove this question has a positive answer if and is a mapping of L p integrable dilatation for p > n − 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence for which Let S (·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences and (where and the sequence ( u k ) has its dilatation pointwise bounded above by an L r integrable function, r >  n − 1) that satisfy as k → ∞ and for which the sign of the det tends to 1 in L 1 . This result contains Reshetnyak’s theorem as the special case ( u k ) ≡ Id , p = 1.