Relaxed energies, defect measures, and minimal currents (Q2164143)

A natural existence question for a continuous harmonic map with a suitably given Dirichlet boundary value or in a given homotopic class (a problem posed by R. Schoen) remains open. The author briefly describes several earlier studies concerning energy minimizing harmonic maps, and maps that minimize the so-called relaxed energy from \(\mathbb{R}^3\) into \(S^2\). Of particular interest is the partial regularity and properties of possible singularities of such maps. A sketch proof of a formula conjectured by \textit{H. Brezis} and \textit{P. Mironescu} [Sobolev maps to the circle. From the perspective of analysis, geometry, and topology. New York, NY: Birkhäuser (2021; Zbl 1501.46001)] is provided, concerning the relaxed \(k\)-energy for Sobolev maps from \(\mathbb{R}^n\) to \(S^k\), for \(k>1\). For the entire collection see [Zbl 1491.46003].