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

  • Katarzyna Mazowiecka und Paweł Strzelecki EMAIL logo
Veröffentlicht/Copyright: 1. Mai 2016

Abstract

We consider the set of smooth zero degree maps ψ:𝕊2𝕊2 which have the following properties: (i) There is a unique minimizing harmonic map u:𝔹3𝕊2 which agrees with ψ on the boundary of the unit ball. (ii) The map u has at least N singular points in 𝔹3. (iii) The Lavrentiev gap phenomenon holds for ψ, i.e., the infimum of the Dirichlet energies E(w) of all smooth extensions w:𝔹3𝕊2 of ψ is strictly larger than the Dirichlet energy 𝔹3|u|2 of the (irregular) minimizer u. For each positive integer N, we prove that this set is dense in the set 𝒮 of all smooth zero degree maps ϕ:𝕊2𝕊2 endowed with the W1,p-topology, where 1 p ¡ 2. This result is sharp since it fails in the W1,2-topology of 𝒮.

MSC 2010: 58E20; 35J57; 35J50

Communicated by Frank Duzaar


Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2012/07/B/ST1/03366

Funding statement: The work of both authors has been partially supported by the Narodowe Centrum Nauki grant no. 2012/07/B/ST1/03366.

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Received: 2015-10-13
Revised: 2016-2-26
Accepted: 2016-4-11
Published Online: 2016-5-1
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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