On double-covering stationary points of a constrained Dirichlet energy

Bevan, Jonathan

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 391-411 / Harvested from Numdam

Résumé

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

Abstract

The double-covering map $ {\mathbf{u}}_{\mathrm{dc}}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{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 $ {\mathbf{u}}_{\mathrm{dc}}$ is the global minimizer of the Dirichlet energy $ I(\mathbf{u})={\int }_{B}{|\mathrm{\nabla }\mathbf{u}|}^{2}\,d\mathbf{x}$ among all $ {W}^{1,2}$ mappings u of the unit ball $ B\subset {\mathbb{R}}^{2}$ satisfying (i) $ \mathbf{u}={\mathbf{u}}_{\mathrm{dc}}$ on ∂B, and (ii) $ \mathrm{det}\,\mathrm{\nabla }\mathbf{u}=1$ almost everywhere. Let the class of such admissible maps be $ \mathcal{A}$. The chief innovation is to express $ I(\mathbf{u})$ in terms of an auxiliary functional $ G(\mathbf{u}-{\mathbf{u}}_{\mathrm{dc}})$, using which we show that $ {\mathbf{u}}_{\mathrm{dc}}$ is a stationary point of I in $ \mathcal{A}$, and that $ {\mathbf{u}}_{\mathrm{dc}}$ is a global minimizer of the Dirichlet energy among members of $ \mathcal{A}$ whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about $ {\mathbf{u}}_{\mathrm{dc}}$ in $ \mathcal{A}$ using ODE techniques, we also show that $ {\mathbf{u}}_{\mathrm{dc}}$ is a local minimizer among variations whose tangent ψ to $ \mathcal{A}$ at $ {\mathbf{u}}_{\mathrm{dc}}$ obeys $ G({\mathit{\psi }}^{\mathrm{o}})>0$, where $ {\mathit{\psi }}^{\mathrm{o}}$ is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint $ \mathrm{det}\,\mathrm{\nabla }\mathbf{u}=1$ is identified by an analysis which exploits the well-known Fefferman–Stein duality.

Détail

Publié le : 2014-01-01

MR 3181676 | Zbl 1311.49009

DOI : https://doi.org/10.1016/j.anihpc.2013.04.001
Classification:  35A15,  49J40,  49N60

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