On double-covering stationary points of a constrained Dirichlet energy
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 391-411
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The double-covering map 𝐮 dc : 2 2 is given by 𝐮 dc (𝐱)=1 2|𝐱|x 2 2 -x 1 2 2x 1 x 2 in cartesian coordinates. This paper examines the conjecture that 𝐮 dc is the global minimizer of the Dirichlet energy I(𝐮)= B |𝐮| 2 d𝐱 among all W 1,2 mappings u of the unit ball B 2 satisfying (i) 𝐮=𝐮 dc on ∂B, and (ii) det 𝐮=1 almost everywhere. Let the class of such admissible maps be 𝒜. The chief innovation is to express I(𝐮) in terms of an auxiliary functional G(𝐮-𝐮 dc ), using which we show that 𝐮 dc is a stationary point of I in 𝒜, and that 𝐮 dc is a global minimizer of the Dirichlet energy among members of 𝒜 whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about 𝐮 dc in 𝒜 using ODE techniques, we also show that 𝐮 dc is a local minimizer among variations whose tangent ψ to 𝒜 at 𝐮 dc obeys G(ψ o )>0, where ψ o is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det 𝐮=1 is identified by an analysis which exploits the well-known Fefferman–Stein duality.

Le double-revêtement 𝐮 dc : 2 2 est donné par 𝐮 dc (𝐱)=1 2|𝐱|x 2 2 -x 1 2 2x 1 x 2 en coordonnées cartésiennes. Cet article examine la conjecture selon laquelle 𝐮 dc est le minimiseur global de l'énergie de Dirichlet I(𝐮)= B |𝐮| 2 d𝐱 pour les fonctions satisfaisant (i) 𝐮W 1,2 (B), où B est la boule unité de 2 , (ii) 𝐮=𝐮 dc sur ∂B, et (iii) det 𝐮=1 presque partout. Soit 𝒜 la classe admissible de telles fonctions. La principale innovation est ici d'exprimer I(𝐮) sous forme d'une fonction auxiliaire G(𝐮-𝐮 dc ), avec laquelle nous montrons que 𝐮 dc est un point stationnaire de I en 𝒜, et que 𝐮 dc est un minimiseur global de l'énergie de Dirichlet parmi les membres de 𝒜 dont la décomposition de Fourier peut être contrôlée d'une manière détaillée dans l'article. En construisant des variations autour de 𝐮 dc en 𝒜 par des techniques variationnelles, nous montrons également que 𝐮 dc est un minimiseur local parmi les variations dont la tangente ψ de 𝐮 dc vers 𝒜 obéissent à G(ψ o )>0, où ψ o est la partie impaire de ψ. Additionnellement, un multiplicateur de Lagrange correspondant à la contrainte det 𝐮=1 est identifié par une analyse qui exploite la dualité de Fefferman–Stein.

DOI : https://doi.org/10.1016/j.anihpc.2013.04.001
Classification:  35A15,  49J40,  49N60
@article{AIHPC_2014__31_2_391_0,
     author = {Bevan, Jonathan},
     title = {On double-covering stationary points of a constrained Dirichlet energy},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     pages = {391-411},
     doi = {10.1016/j.anihpc.2013.04.001},
     zbl = {1311.49009},
     mrnumber = {3181676},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_2_391_0}
}
Bevan, Jonathan. On double-covering stationary points of a constrained Dirichlet energy. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 391-411. doi : 10.1016/j.anihpc.2013.04.001. http://www.numdam.org/item/AIHPC_2014__31_2_391_0/

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