We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{ S}^2\to \mathbb{ S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{ B}^3\to \mathbb{ S}^2$ 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 $w\colon \mathbb{ B}^3\to\mathbb{ S}^2$ 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 $\phi\colon \mathbb{ S}^2\to\mathbb{ S}^2$ 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.
Categories: math.AP