On double-covering stationary points of a constrained Dirichlet energy
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 391-411.

Le double-revêtement ${𝐮}_{\mathrm{dc}}:{ℝ}^{2}\to {ℝ}^{2}$ est donné par

 ${𝐮}_{\mathrm{dc}}\left(𝐱\right)=\frac{1}{\sqrt{2}|𝐱|}\left(\begin{array}{c}\hfill {x}_{2}^{2}-{x}_{1}^{2}\hfill \\ \hfill 2{x}_{1}{x}_{2}\hfill \end{array}\right)$
en coordonnées cartésiennes. Cet article examine la conjecture selon laquelle ${𝐮}_{\mathrm{dc}}$ est le minimiseur global de l'énergie de Dirichlet $I\left(𝐮\right)={\int }_{B}{|\nabla 𝐮|}^{2}\phantom{\rule{0.166667em}{0ex}}d𝐱$ pour les fonctions satisfaisant (i) $𝐮\in {W}^{1,2}\left(B\right)$, où B est la boule unité de ${ℝ}^{2}$, (ii) $𝐮={𝐮}_{\mathrm{dc}}$ sur ∂B, et (iii) $\mathrm{det}\phantom{\rule{0.166667em}{0ex}}\nabla 𝐮=1$ presque partout. Soit $𝒜$ la classe admissible de telles fonctions. La principale innovation est ici d'exprimer $I\left(𝐮\right)$ sous forme d'une fonction auxiliaire $G\left(𝐮-{𝐮}_{\mathrm{dc}}\right)$, avec laquelle nous montrons que ${𝐮}_{\mathrm{dc}}$ est un point stationnaire de I en $𝒜$, et que ${𝐮}_{\mathrm{dc}}$ 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 ${𝐮}_{\mathrm{dc}}$ en $𝒜$ par des techniques variationnelles, nous montrons également que ${𝐮}_{\mathrm{dc}}$ est un minimiseur local parmi les variations dont la tangente ψ de ${𝐮}_{\mathrm{dc}}$ vers $𝒜$ obéissent à $G\left({\psi }^{\mathrm{o}}\right)>0$, où ${\psi }^{\mathrm{o}}$ est la partie impaire de ψ. Additionnellement, un multiplicateur de Lagrange correspondant à la contrainte $\mathrm{det}\phantom{\rule{0.166667em}{0ex}}\nabla 𝐮=1$ est identifié par une analyse qui exploite la dualité de Fefferman–Stein.

The double-covering map ${𝐮}_{\mathrm{dc}}:{ℝ}^{2}\to {ℝ}^{2}$ is given by

 ${𝐮}_{\mathrm{dc}}\left(𝐱\right)=\frac{1}{\sqrt{2}|𝐱|}\left(\begin{array}{c}\hfill {x}_{2}^{2}-{x}_{1}^{2}\hfill \\ \hfill 2{x}_{1}{x}_{2}\hfill \end{array}\right)$
in cartesian coordinates. This paper examines the conjecture that ${𝐮}_{\mathrm{dc}}$ is the global minimizer of the Dirichlet energy $I\left(𝐮\right)={\int }_{B}{|\nabla 𝐮|}^{2}\phantom{\rule{0.166667em}{0ex}}d𝐱$ among all ${W}^{1,2}$ mappings u of the unit ball $B\subset {ℝ}^{2}$ satisfying (i) $𝐮={𝐮}_{\mathrm{dc}}$ on ∂B, and (ii) $\mathrm{det}\phantom{\rule{0.166667em}{0ex}}\nabla 𝐮=1$ almost everywhere. Let the class of such admissible maps be $𝒜$. The chief innovation is to express $I\left(𝐮\right)$ in terms of an auxiliary functional $G\left(𝐮-{𝐮}_{\mathrm{dc}}\right)$, using which we show that ${𝐮}_{\mathrm{dc}}$ is a stationary point of I in $𝒜$, and that ${𝐮}_{\mathrm{dc}}$ 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 ${𝐮}_{\mathrm{dc}}$ in $𝒜$ using ODE techniques, we also show that ${𝐮}_{\mathrm{dc}}$ is a local minimizer among variations whose tangent ψ to $𝒜$ at ${𝐮}_{\mathrm{dc}}$ obeys $G\left({\psi }^{\mathrm{o}}\right)>0$, where ${\psi }^{\mathrm{o}}$ is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint $\mathrm{det}\phantom{\rule{0.166667em}{0ex}}\nabla 𝐮=1$ is identified by an analysis which exploits the well-known Fefferman–Stein duality.

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},
pages = {391--411},
publisher = {Elsevier},
volume = {31},
number = {2},
year = {2014},
doi = {10.1016/j.anihpc.2013.04.001},
zbl = {1311.49009},
mrnumber = {3181676},
language = {en},
url = {http://archive.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, Tome 31 (2014) no. 2, pp. 391-411. doi : 10.1016/j.anihpc.2013.04.001. http://archive.numdam.org/item/AIHPC_2014__31_2_391_0/

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