Convex integration with linear constraints and its applications

Abstract

We study solutions of the first order partial differential inclusions of the form ∇u ∈ K, where u : Ω ⊂ R n → R m and K is a set of m × n real matrices, and derive a companion version to the result of Müller andŠverák [20], concerning a general linear constraint on the components of ∇u. We then consider two applications: the vectorial eikonal equation and a T4-configuration both under linear constraints.2010 Mathematics Subject Classification. 35F60, 35D30.