On double-covering stationary points of a constrained Dirichlet energy

Bevan, Jonathan

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 391-411

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

$${𝐮}_{\mathrm{dc}}\left(𝐱\right)=\frac{1}{\sqrt{2}\left|𝐱\right|}\left(\begin{array}{c}\hfill x_2^2-x_1^2\hfill \\ \hfill 2x_1{x}_2\hfill \end{array}\right)$$

in cartesian coordinates. This paper examines the conjecture that ${𝐮}_{\mathrm{dc}}$ is the global minimizer of the Dirichlet energy $I\left(𝐮\right)={\int }_B{\left|\nabla 𝐮\right|}^{2}d𝐱$ 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) $\det \nabla 𝐮=1$ 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 $G(𝐮-{𝐮}_{\mathrm{dc}})$, 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 $\det \nabla 𝐮=1$ is identified by an analysis which exploits the well-known Fefferman–Stein duality.

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

$${𝐮}_{\mathrm{dc}}\left(𝐱\right)=\frac{1}{\sqrt{2}\left|𝐱\right|}\left(\begin{array}{c}\hfill x_2^2-x_1^2\hfill \\ \hfill 2x_1{x}_2\hfill \end{array}\right)$$

en coordonnées cartésiennes. Cet article examine la conjecture selon laquelle ${𝐮}_{\mathrm{dc}}$ est le minimiseur global de l'énergie de Dirichlet $I\left(𝐮\right)={\int }_B{\left|\nabla 𝐮\right|}^{2}d𝐱$ 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) $\det \nabla 𝐮=1$ 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 $G(𝐮-{𝐮}_{\mathrm{dc}})$, 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 $\det \nabla 𝐮=1$ est identifié par une analyse qui exploite la dualité de Fefferman–Stein.

Classification : 35A15, 49J40, 49N60

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