no. 2
Existence of p harmonic multiple valued maps into a separable Hilbert space
[Existence d’applications p harmoniques multivaluées dans un espace de Hilbert séparable]
Bouafia, Philippe1 ; De Pauw, Thierry2 ; Goblet, Jordan3
1 ENSEA 6 avenue du Ponceau 95014 Cergy-Pontoise Cdex (France)
2 Institut de Mathmatiques de Jussieu - PRG UMR 7586 Equipe Gomtrie et Dynamique Btiment Sophie Germain Case 7012 75205 Paris cedex 13 (France)
3 Institut de recherche en mathmatique et physique Chemin du cyclotron, 2 1348 Louvain-la-Neuve (Belgique)
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.

DOI : 10.5802/aif.2944
Classification : 49Q20, 35J50
Keywords: Multiple valued maps, $p$ harmonic
Mot clés : Applications multivaluées, $p$ harmonique

Bouafia, Philippe 1 ; De Pauw, Thierry 2 ; Goblet, Jordan 3

1 ENSEA 6 avenue du Ponceau 95014 Cergy-Pontoise Cdex (France)
2 Institut de Mathmatiques de Jussieu - PRG UMR 7586 Equipe Gomtrie et Dynamique Btiment Sophie Germain Case 7012 75205 Paris cedex 13 (France)
3 Institut de recherche en mathmatique et physique Chemin du cyclotron, 2 1348 Louvain-la-Neuve (Belgique) 
@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},
     pages = {763--833},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     doi = {10.5802/aif.2944},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2944/}
}
TY  - JOUR
AU  - Bouafia, Philippe
AU  - De Pauw, Thierry
AU  - Goblet, Jordan
TI  - Existence of $p$ harmonic multiple valued maps into a separable Hilbert space
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 763
EP  - 833
VL  - 65
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2944/
DO  - 10.5802/aif.2944
LA  - en
ID  - AIF_2015__65_2_763_0
ER  - 
%0 Journal Article
%A Bouafia, Philippe
%A De Pauw, Thierry
%A Goblet, Jordan
%T Existence of $p$ harmonic multiple valued maps into a separable Hilbert space
%J Annales de l'Institut Fourier
%D 2015
%P 763-833
%V 65
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2944/
%R 10.5802/aif.2944
%G en
%F 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) no. 2, pp. 763-833. doi : 10.5802/aif.2944. https://aif.centre-mersenne.org/articles/10.5802/aif.2944/

[1] Almgren, F. Deformations and multiple-valued functions, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) (Proc. Sympos. Pure Math.), Volume 44, Amer. Math. Soc., Providence, RI, 1986, pp. 29-130 | DOI | MR | Zbl

[2] Almgren, Frederick J. Jr. Almgren’s big regularity paper, World Scientific Monograph Series in Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000, pp. xvi+955 (Q-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

[3] Benyamini, Yoav; Lindenstrauss, Joram Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, 2000, pp. xii+488 | MR | Zbl

[4] Brink, David Hölder continuity of roots of complex and p-adic polynomials, Comm. Algebra, Volume 38 (2010) no. 5, pp. 1658-1662 | DOI | MR | Zbl

[5] Dacorogna, Bernard Direct methods in the calculus of variations, Applied Mathematical Sciences, 78, Springer-Verlag, Berlin, 1989, pp. x+308 | DOI | MR | Zbl

[6] De Lellis, Camillo; Spadaro, Emanuele Nunzio Q-valued functions revisited, Mem. Amer. Math. Soc., Volume 211 (2011) no. 991, pp. vi+79 | DOI | MR | Zbl

[7] Diestel, J.; Uhl, J. J. Jr. Vector measures, American Mathematical Society, Providence, R.I., 1977, pp. xiii+322 (With a foreword by B. J. Pettis, Mathematical Surveys, No. 15) | MR | Zbl

[8] Edwards, R. E. Functional analysis, Dover Publications, Inc., New York, 1995, pp. xvi+783 (Theory and applications, Corrected reprint of the 1965 original) | MR | Zbl

[9] Federer, Herbert Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, pp. xiv+676 | MR | Zbl

[10] Goblet, Jordan A selection theory for multiple-valued functions in the sense of Almgren, Ann. Acad. Sci. Fenn. Math., Volume 31 (2006) no. 2, pp. 297-314 | MR | Zbl

[11] Goblet, Jordan Lipschitz extension of multiple Banach-valued functions in the sense of Almgren, Houston J. Math., Volume 35 (2009) no. 1, pp. 223-231 | MR | Zbl

[12] Johnson, William B.; Lindenstrauss, Joram; Schechtman, Gideon Extensions of Lipschitz maps into Banach spaces, Israel J. Math., Volume 54 (1986) no. 2, pp. 129-138 | DOI | MR | Zbl

[13] Lang, Urs; Schlichenmaier, Thilo Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. (2005) no. 58, pp. 3625-3655 | DOI | MR | Zbl

[14] Malý, Jan; Ziemer, William P. Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, 51, American Mathematical Society, Providence, RI, 1997, pp. xiv+291 | DOI | MR | Zbl

[15] Marden, Morris Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966, pp. xiii+243 | MR | Zbl

[16] White, B. (Personal communication)

[17] Whitney, Hassler Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934) no. 1, pp. 63-89 | DOI | MR | Zbl

[18] Whitney, Hassler Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972, pp. xii+399 | MR | Zbl

Cité par Sources :