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

We consider the vectorial system $$\{\begin{array}{lll}Du\in \mathcal{O}(2),& a\mathrm{.}e\mathrm{.}in\mathrm{\Omega },u=0,& on\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a subset of ${\mathbb{R}}^2$, $u:\mathrm{\Omega }\to {\mathbb{R}}^2$ 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.