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

2014/06/03 by Katarzyna Mazowiecka, Mazowiecka, Katarzyna, Paweł Strzelecki +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1406.0601

openalex publication_date 2014/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for each positive integer N the set of smooth, zero degree\nmaps \ψ colon mathbbS²\→ mathbbS² which have the following three\nproperties:\n (1) there is a unique minimizing harmonic map u colon mathbbB³\→\n mathbbS² which agrees with \ψ on the boundary of the unit ball;\n (2) this map u has at least N singular points in mathbbB³;\n (3) the Lavrentiev gap phenomenon holds for \ψ, i.e., the infimum of the\nDirichlet energies E(w) of all smooth extensions w colon\n mathbbB³\→ mathbbS² of \ψ is strictly larger than the Dirichlet\nenergy \∫_ mathbbB³ |\∇ u|² of the (irregular) minimizer u, is\ndense in the set of all smooth zero degree maps \φ colon\n mathbbS²\→ mathbbS² endowed with the W^1,p-topology, where 1\≤ p\n< 2. This result is sharp: it fails in the W^1,2 topology on the set of\nall smooth boundary data.\n

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