Density of smooth maps for fractional Sobolev spaces W s,p into simply connected manifolds when s1
Confluentes Mathematici, Volume 5 (2013) no. 2, p. 3-24

Given a compact manifold N n ν and real numbers s1 and 1p<, we prove that the class C (Q ¯ m ;N n ) of smooth maps on the cube with values into N n is strongly dense in the fractional Sobolev space W s,p (Q m ;N n ) when N n is sp simply connected. For sp integer, we prove weak sequential density of C (Q ¯ m ;N n ) when N n is sp-1 simply connected. The proofs are based on the existence of a retraction of ν onto N n except for a small subset of N n and on a pointwise estimate of fractional derivatives of composition of maps in W s,p W 1,sp .

Received : 2012-08-10
Revised : 2013-03-05
Accepted : 2013-03-05
Published online : 2017-03-27
Classification:  58D15,  46E35,  46T20
Keywords: Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness
@article{CML_2013__5_2_3_0,
     author = {Pierre Bousquet and Augusto C. Ponce and Jean Van Schaftingen},
     title = {Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {5},
     number = {2},
     year = {2013},
     pages = {3-24},
     language = {en},
     url = {https://cml.centre-mersenne.org/item/CML_2013__5_2_3_0}
}
Bousquet, Pierre; Ponce, Augusto C.; Van Schaftingen, Jean. Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$. Confluentes Mathematici, Volume 5 (2013) no. 2, pp. 3-24. cml.centre-mersenne.org/item/CML_2013__5_2_3_0/

[1] Robert A. Adams Sobolev spaces, Academic Press, New York-London (Pure and Applied Mathematics, Vol. 65)

[2] Fabrice Bethuel A characterization of maps in H 1 (B 3 ,S 2 ) which can be approximated by smooth maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 7, pp. 269-286

[3] Fabrice Bethuel Approximations in trace spaces defined between manifolds, Nonlinear Anal., Tome 24 no. 1, pp. 121-130 | Article

[4] Fabrice Bethuel The approximation problem for Sobolev maps between two manifolds, Acta Math., Tome 167 no. 3-4, pp. 153-206 | Article

[5] Haïm Brezis; Petru Mironescu (in preparation)

[6] Haïm Brezis; Petru Mironescu Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., Tome 1 no. 4, pp. 387-404 | Article

[7] Haïm Brezis; Louis Nirenberg Degree theory and BMO, Part I : compact manifolds without boundaries, Selecta Math., pp. 197-263

[8] Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen Strong density for higher order Sobolev spaces into compact manifolds (submitted paper)

[9] Fabrice Bethuel; Xiao Min Zheng Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., Tome 80 no. 1, pp. 60-75 | Article

[10] Miguel Escobedo Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions, Rev. Mat. Univ. Complut. Madrid, Tome 1 no. 1-3, pp. 127-144

[11] Herbert Federer; Wendell H. Fleming Normal and integral currents, Ann. of Math. (2), Tome 72, pp. 458-520

[12] Emilio Gagliardo Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, Tome 27, pp. 284-305

[13] Emilio Gagliardo Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., Tome 8, pp. 24-51

[14] Andreas Gastel; Andreas J. Nerf Density of smooth maps in W k,p (M,N) for a close to critical domain dimension, Ann. Global Anal. Geom., Tome 39 no. 2, pp. 107-129

[15] Piotr Hajłasz Approximation of Sobolev mappings, Nonlinear Anal., Tome 22 no. 12, pp. 1579-1591 | Article

[16] Fengbo Hang Density problems for W 1,1 (M,N), Comm. Pure Appl. Math., Tome 55 no. 7, pp. 937-947 | Article

[17] Lars Inge Hedberg On certain convolution inequalities, Proc. Amer. Math. Soc., Tome 36, pp. 505-510

[18] Robert Hardt; David Kinderlehrer; Fang-Hua Lin Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 5 no. 4, pp. 297-322

[19] Fengbo Hang; Fanghua Lin Topology of Sobolev mappings. II, Acta Math., Tome 191 no. 1, pp. 55-107

[20] Fengbo Hang; Fanghua Lin Topology of Sobolev mappings. III, Comm. Pure Appl. Math., Tome 56 no. 10, pp. 1383-1415 | Article

[21] Vladimir Mazʼya Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, Tome 342, Springer, xxviii+866 pages | Article

[22] Petru Mironescu Sobolev maps on manifolds: degree, approximation, lifting, Perspectives in nonlinear partial differential equations (Contemp. Math.) Tome 446, pp. 413-436 (In honor of Haïm Brezis) | Article

[23] Vladimir Mazʼya; Tatyana Shaposhnikova An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces, J. Evol. Equ., Tome 2 no. 1, pp. 113-125 | Article

[24] Domenico Mucci Strong density results in trace spaces of maps between manifolds, Manuscripta Math., Tome 128 no. 4, pp. 421-441 | Article

[25] Louis Nirenberg On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), Tome 13, pp. 115-162

[26] Frédérique Oru Rôle des oscillations dans quelques problèmes d’analyse non linéaire (Thèse de doctorat)

[27] Mohammad Reza Pakzad Weak density of smooth maps in W 1,1 (M,N) for non-abelian π 1 (N), Ann. Global Anal. Geom., Tome 23 no. 1, pp. 1-12 | Article

[28] Mohammad Reza Pakzad; Tristan Rivière Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal., Tome 13 no. 1, pp. 223-257 | Article

[29] Tristan Rivière Dense subsets of H 1/2 (S 2 ,S 1 ), Ann. Global Anal. Geom., Tome 18 no. 5, pp. 517-528 | Article

[30] Thomas Runst; Winfried Sickel Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, Tome 3, Walter de Gruyter & Co., x+547 pages | Article

[31] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press

[32] Richard Schoen; Karen Uhlenbeck Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., Tome 18 no. 2, pp. 253-268

[33] Brian White Infima of energy functionals in homotopy classes of mappings, J. Differential Geom., Tome 23 no. 2, pp. 127-142