Convex integration with linear constraints and its applications

Abstract

We study solutions of the first order partial differential inclusions of the form $\nabla u\in K$, where $u:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ and $K$ is a set of $m\times n$ real matrices, and derive a companion version to the result of {Müller and Šverák} [20], concerning a general linear constraint on the components of $\nabla u$. We then consider two applications: the vectorial eikonal equation and a $T_4$-configuration both under linear constraints.

37 pages