Regularity of minimizing maps with values in \(S^2\) and some numerical simulations (Q2733856)

The authors consider the minimization problem NEWLINE\[NEWLINE\inf\int_\Omega|\nabla u|^2+ \lambda \int_\Omega|u-f|^2NEWLINE\]NEWLINE among mappings \(u\in H^1(\Omega, S^2)\), where \(\Omega\) is the unit ball in \(\mathbb{R}^3\) with boundary \(S^2\). Here \(f\) denotes a measurable function from \(\Omega\) into \(S^2\) and \(\lambda\geq 0\) is a parameter. The main result states that minimizers are regular in the case \(0\leq \lambda<{3\over 5}\). This improves an earlier result of \textit{R. Hadiji} and \textit{F. Zhou} [Potential Anal. 1, No. 4, 385-400 (1992; Zbl 0794.35036)] where additional restrictions were imposed on \(f\). However, the paper describes an algorithm converging to an exact solution.