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

Abstract

We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), & \mbox{a.e. in}\;Ω, u=0, & \mbox{on} \;\partial Ω, \end{cases} \] where $\Omega$ is a subset of ${\mathbb{R}}^2$, $u:\Omega \to {\mathbb{R}}^2$ and $\mathcal{O}(2)$ is the orthogonal group of ${\mathbb{R}}^2$. 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.