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- W1515692906 abstract "We prove that for each positive integer $N$ the set of smooth, zero degree maps $psicolonmathbb{S}^2to mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $ucolon mathbb{B}^3to mathbb{S}^2$ 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 $wcolon mathbb{B}^3tomathbb{S}^2$ 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 $phicolon mathbb{S}^2tomathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1le p < 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data." @default.
- W1515692906 created "2016-06-24" @default.
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- W1515692906 date "2014-06-03" @default.
- W1515692906 modified "2023-09-27" @default.
- W1515692906 title "The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data" @default.
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