Minimization of conformally invariant energies in homotopy classes
Let \(X\subset \mathbb{R}^N\) be compact and \(\alpha\in \Pi_n(X)\) a nontrivial homotopy class of maps \(u: S^n\to X\). The authors analyze the behavior of a minimizing sequence \(u_k\in\alpha\) for a conformally invariant functional such as the \(n\)-energy \[ E(u)= \int_{S^n}| du(\xi)|^nd\xi. \] A collection \((u^i)_{i\in I}\) of nonconstant maps is a \textit{weak limit set} of a sequence \(u_k\) if there exist mutually unbounded conformal automorphisms \(g^i_k\) of the sphere such that \(u_k\circ g^i_k\) converges to \(u^i\) weakly for any \(i\in I\). If in addition the sum of the energies of the \(u^i\) equals the limit of energy along the sequence, the convergence is said to be \textit{in energy}. The main results follow from compactness and continuity results formulated in terms of weak limit set and convergence in energy and the reasoning follows the direct method of calculus of variations.
scientific article; zbMATH DE number 1160791