A minimization problem associated with gap phenomenon (Q1373815)

We consider the minimization problem \[ \inf\left\{ \int_\Omega|D(u-f)|^2: u\in C^1(\overline{\Omega}, S^2) \right\},\tag{1} \] assuming that \(f\) satisfies certain conditions. We develop a method to prove that for any weak solution of the Euler equation of (1), we have \(\text{Sing}(f) \subset \text{Sing}(u)\), where \(\text{Sing}(u)\) is the singular set of \(u\). As a direct corollary of this result, (1) is not achieved.