Variational methods for the selection of solutions to an implicit system of PDEs

TL;DR

This paper introduces a variational approach to select specific solutions to a PDE system involving orthogonal matrices by minimizing a weighted measure of the solution's singular set.

Contribution

It develops a novel variational framework to distinguish solutions of an implicit PDE system based on the singularity measure of their gradients.

Findings

Identifies a variational criterion for solution selection.

Provides a method to minimize the singular set measure.

Offers insights into the structure of solutions to the PDE system.

Abstract

We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), & \mbox{a.e. in}\;\Omega, u=0, & \mbox{on} \;\partial \Omega, \end{cases} \] where $\Omega$ is a subset of $\R^2$, $u:\Omega\to \R^2$ and $\mathcal{O}(2)$ is the orthogonal group of $\R^2$. We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appropriate weighted measure of the singular set of the gradient.