A problem of minimization with relaxed energy
Let \(\Omega\) denote the unit ball in \(\mathbb{R}^3\) and \(S^2\) the unit sphere and let \[ E(u)=\int_\Omega |\nabla u|^2dxdydz. \] The authors consider the minimization problem \[ \alpha =\inf\{E(u-f),\quad u\in C^1(\overline{\Omega},S^2)\} \] where \(f\) is a \(H^1\) map from \(\Omega\) to \(\mathbb{R}^3\) which has values almost everywhere in \(S^2\) and which is regular except at a finite number of points. They generalize the result of \textit{F. Bethuel} and \textit{H. Brezis} [C.R. Acad. Sci., Paris, Sér. I 310, No. 13, 859-864 (1990; Zbl 0717.49005)] and show that the infimum is not attained and that every minimizing sequence converges strongly in \(H^1\) to a regular mapping of degree one from \(S^2\) to \(S^2\).
scientific article; zbMATH DE number 942769
scientific article