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

Abstract

We consider the vectorial system $$\begin{aligned} {\left\{ \begin{array}{ll} Du \in \mathcal {O}(2), &{} \text{ a.e. } \text{ in }\,\;\Omega , u=0, &{} \text{ on } \,\;\partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a subset of $$\mathbb R^2$$ , $$u:\Omega \rightarrow \mathbb R^2$$ 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 some set of singularities of the gradient.