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 a.e. in Ω, where u is a (Lipschitz) mapping of an open set Ω ⊂ Rn into Rm , ∇u(x) is its gradient (i.e. 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)...