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 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 harmoniques de 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 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 harmonic Hilbert space multiple valued maps of variables.
@article{AIF_2015__65_2_763_0,
author = {Bouafia, Philippe and De Pauw, Thierry and Goblet, Jordan},
title = {Existence of $p$ harmonic multiple valued maps into a separable Hilbert space},
journal = {Annales de l'Institut Fourier},
volume = {65},
year = {2015},
pages = {763-833},
doi = {10.5802/aif.2944},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2015__65_2_763_0}
}
Bouafia, Philippe; De Pauw, Thierry; Goblet, Jordan. Existence of $p$ harmonic multiple valued maps into a separable Hilbert space. Annales de l'Institut Fourier, Tome 65 (2015) pp. 763-833. doi : 10.5802/aif.2944. http://gdmltest.u-ga.fr/item/AIF_2015__65_2_763_0/
[1] Deformations and multiple-valued functions, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 44 (1986), pp. 29-130 | Article | MR 840266 | Zbl 0595.49028
[2] Almgren’s big regularity paper, World Scientific Publishing Co., Inc., River Edge, NJ, World Scientific Monograph Series in Mathematics, Tome 1 (2000), pp. xvi+955 (-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer) | Zbl 0985.49001
[3] Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 48 (2000), pp. xii+488 | MR 1727673 | Zbl 0946.46002
[4] Hölder continuity of roots of complex and -adic polynomials, Comm. Algebra, Tome 38 (2010) no. 5, pp. 1658-1662 | Article | MR 2642016 | Zbl 1213.12001
[5] Direct methods in the calculus of variations, Springer-Verlag, Berlin, Applied Mathematical Sciences, Tome 78 (1989), pp. x+308 | Article | MR 990890 | Zbl 0703.49001
[6] -valued functions revisited, Mem. Amer. Math. Soc., Tome 211 (2011) no. 991, pp. vi+79 | Article | MR 2663735 | Zbl 1246.49001
[7] Vector measures, American Mathematical Society, Providence, R.I. (1977), pp. xiii+322 (With a foreword by B. J. Pettis, Mathematical Surveys, No. 15) | MR 453964 | Zbl 0369.46039
[8] Functional analysis, Dover Publications, Inc., New York (1995), pp. xvi+783 (Theory and applications, Corrected reprint of the 1965 original) | MR 1320261 | Zbl 0182.16101
[9] Geometric measure theory, Springer-Verlag New York Inc., New York, Die Grundlehren der mathematischen Wissenschaften, Band 153 (1969), pp. xiv+676 | MR 257325 | Zbl 0176.00801
[10] A selection theory for multiple-valued functions in the sense of Almgren, Ann. Acad. Sci. Fenn. Math., Tome 31 (2006) no. 2, pp. 297-314 | MR 2248817 | Zbl 1143.54009
[11] Lipschitz extension of multiple Banach-valued functions in the sense of Almgren, Houston J. Math., Tome 35 (2009) no. 1, pp. 223-231 | MR 2491878 | Zbl 1165.54004
[12] Extensions of Lipschitz maps into Banach spaces, Israel J. Math., Tome 54 (1986) no. 2, pp. 129-138 | Article | MR 852474 | Zbl 0626.46007
[13] Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. (2005) no. 58, pp. 3625-3655 | Article | MR 2200122 | Zbl 1095.53033
[14] Fine regularity of solutions of elliptic partial differential equations, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 51 (1997), pp. xiv+291 | Article | MR 1461542 | Zbl 0882.35001
[15] Geometry of polynomials, American Mathematical Society, Providence, R.I., Second edition. Mathematical Surveys, No. 3 (1966), pp. xiii+243 | MR 225972 | Zbl 0162.37101
[16] (Personal communication)
[17] Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Tome 36 (1934) no. 1, pp. 63-89 | Article | MR 1501735 | Zbl 0008.24902
[18] Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1972), pp. xii+399 | MR 387634 | Zbl 0265.32008