Existence of $p$ harmonic multiple valued maps into a separable Hilbert space

Bouafia, Philippe; De Pauw, Thierry; Goblet, Jordan

Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 763-833.

Nous étudions les propriétés élémentaires d’applications multivaluées entre espaces métriques : mesurabilité, intégrabilité, continuité, caractère lipschitzien, extension lipschitzienne, et différentiabilité dans le cas d’espaces vectoriels. Nous rappelons le théorème de plongement de F.J. Almgren et nous démontrons un nouveau théorème de plongement, plus général, dont on déduit ensuite un théorème de compacité à la Fréchet-Kolmogoroff pour les espaces $L_p$ d’applications multivaluées. Nous introduisons une définition intrinsèque d’applications de Sobolev multivaluées à valeurs dans un espace de Hilbert et nous développons les outils classiques dans ce cadre : extension de Sobolev, inégalité de Poincaré, approximation de type Lusin par des applications lipschitziennes, théorie de trace, et l’analogue du théorème de compacité de Rellich. Nous obtenons en corollaire un résultat d’existence pour le problème de Dirichlet des applications multivaluées $p$ harmoniques de $m$ variables à valeurs dans un espace de Hilbert séparable.

We study the elementary properties of multiple valued maps between two metric spaces: their measurability, Lebesgue integrability, continuity, Lipschitz continuity, Lipschitz extension, and differentiability in case the range and domain are linear. We discuss F.J. Almgren’s embedding Theorem and we prove a new, more general, embedding from which a Fréchet-Kolmogorov compactness Theorem ensues for multiple valued $L_p$ spaces. In turn, we introduce an intrinsic definition of Sobolev multiple valued maps into Hilbert spaces, together with the relevant Sobolev extension property, Poincaré inequality, Luzin type approximation by Lipschitz maps, trace theory, and the analog of Rellich compactness. As a corollary we obtain an existence result for the Dirichlet problem of $p$ harmonic Hilbert space multiple valued maps of $m$ variables.

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