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

TL;DR

This paper demonstrates that for a dense set of smooth zero degree boundary maps into spheres, the Lavrentiev gap phenomenon occurs, with the set being dense in the $W^{1,p}$ topology for $1 \\le p < 2$, but not in $W^{1,2}$.

Contribution

It establishes the density of boundary data exhibiting the Lavrentiev gap phenomenon in the $W^{1,p}$ topology for $p<2$, revealing the phenomenon's sharpness and boundary regularity dependence.

Findings

Lavrentiev gap phenomenon occurs densely for certain boundary maps.

The phenomenon is dense in $W^{1,p}$ for $p<2$ but not in $W^{1,2}$.

The set of boundary maps with at least N singularities is dense in the smooth zero degree maps.

Abstract

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.