TAILIEUCHUNG - Đề tài " Convex integration for Lipschitz mappings and counterexamples to regularity "

In this paper we study Lipschitz solutions of partial differential relations of the form (1) ∇u(x) ∈ K . in Ω, where u is a (Lipschitz) mapping of an open set Ω ⊂ Rn into Rm , ∇u(x) is its gradient (. the matrix ∂ui (x)/∂xj , 1 ≤ i ≤ m, 1 ≤ j ≤ n, defined for almost every x ∈ Ω), and K is a subset of the set M m×n of all real m × n matrices. In addition to relation (1), boundary conditions and other conditions on u will also be considered. Relation (1). | Annals of Mathematics Convex integration for Lipschitz mappings and counterexamples to regularity By S. M uller and V. Sver ak Annals of Mathematics 157 2003 715 742 Convex integration for Lipschitz mappings and counterexamples to regularity By S. Muller and V. SvERAK 1. Introduction In this paper we study Lipschitz solutions of partial differential relations of the form 1 Vu x G K . in Q where u is a Lipschitz mapping of an open set Q c R into Rm Vu x is its gradient . the matrix dui x ỜXj 1 i m 1 j n defined for almost every x G Q and K is a subset of the set Mmxn of all real m X n matrices. In addition to relation 1 boundary conditions and other conditions on u will also be considered. Relation 1 is a special case of partial differential relations which have been extensively studied in connection with certain geometrical problems such as isometric immersions. For example the celebrated results of Nash Na 54 and Kuiper Ku 55 and their far-reaching generalizations by Gromov Gr 86 showed striking and completely unexpected features of the behavior of C 1-isometric immersions of R to R 1 and Lipschitz isometric immersions of R to R . A general result describing a large class of Lipschitz solutions of partial differential relations more general than 1 can be found in the book of Gromov Gr 86 p. 218 . More recently problems concerning solutions of relations of the form 1 have been studied in connection with the characterization of absolute mini-mizers of variational integrals describing the elastic energy of crystals exhibiting interesting microstructures BJ 87 CK 88 . An important observation which came from this direction Ba 90 is that relation 1 can have highly oscillatory solutions even when the difference of any two nonidentical matrices in K has rank 2. This situation which does occur in some very interesting cases is not covered by the theorem of Gromov mentioned above. In technical terms to be explained below the reason is that Gromov s P-convex hull of .

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