The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

Abstract

We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi :{\mathbb{S}}^2\to {\mathbb{S}}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u:{\mathbb{B}}^3\to {\mathbb{S}}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in ${\mathbb{B}}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w:{\mathbb{B}}^3\to {\mathbb{S}}^2$ of $\psi$ is strictly larger than the Dirichlet energy ${\int }_{{\mathbb{B}}^3}|\nabla u{|}^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $\varphi :{\mathbb{S}}^2\to {\mathbb{S}}^2$ endowed with the $W^{1,p}$-topology, where $1\le p<2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data.

arXiv comment14 pages, 3 figures; minor typos etc. corrected