On minimizing $W^{s,1/s}$-maps between circles

Abstract

For $s\in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for ${\mathbb{S}}^1\to {\mathbb{S}}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.