Convex integration with linear constraints and its applications

TL;DR

This paper extends convex integration techniques to include linear constraints on the gradient, providing new solutions and applications to problems like the vectorial eikonal equation and T4-configurations.

Contribution

It introduces a generalized convex integration framework accommodating linear constraints on gradients, expanding the scope of solutions for PDE inclusions.

Findings

Derived a convex integration method with linear constraints.

Constructed solutions for the vectorial eikonal equation under constraints.

Analyzed T4-configuration problems with linear restrictions.

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 {M\"uller and \v{S}ver\'ak} [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.