Relaxed energies, defect measures, and minimal currents

Abstract

In this chapter, we describe very briefly several earlier studies concerning energy minimizing harmonic maps and maps that minimize the so-called relaxed energy from $R^{3}$ into $S^{2}$. Of particular interest is the partial regularity and properties of possible singularities of such maps. We also sketch a proof of a formula conjectured by H. Brezis and P. Mironescu (2021) concerning the relaxed $k$-energy for Sobolev maps from $R^{n}$ into $S^{k}$, for $k>1$.