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Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen
Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$
Confluentes Mathematici, 5 no. 2 (2013), p. 3-24, doi: 10.5802/cml.5
Article PDF | Analyses MR 3145030
Class. Math.: 58D15, 46E35, 46T20
Mots clés: Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness

Résumé - Abstract

Given a compact manifold $N^n \subset {\mathbb{R}}^\nu $ and real numbers $s \ge 1$ and $1 \le p < \infty $, we prove that the class $C^\infty (\overline{Q}^m; N^n)$ 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 $\lfloor sp\rfloor $ simply connected. For $sp$ integer, we prove weak sequential density of $C^\infty (\overline{Q}^m; N^n)$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of ${\mathbb{R}}^\nu $ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.

Bibliographie

[1] Robert A. Adams, Sobolev spaces, Academic Press, New York-London, Pure and Applied Mathematics, Vol. 65  MR 450957 |  Zbl 0314.46030
[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 7, p. 269-286 Numdam |  MR 1067776 |  Zbl 0708.58004
[3] Fabrice Bethuel, “Approximations in trace spaces defined between manifolds”, Nonlinear Anal. 24 no. 1, p. 121-130 Article |  MR 1308474 |  Zbl 0824.58011
[4] Fabrice Bethuel, “The approximation problem for Sobolev maps between two manifolds”, Acta Math. 167 no. 3-4, p. 153-206 Article |  MR 1120602 |  Zbl 0756.46017
[5] Haïm Brezis & Petru Mironescu, “”, in preparation
[6] Haïm Brezis & Petru Mironescu, “Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces”, J. Evol. Equ. 1 no. 4, p. 387-404 Article |  MR 1877265 |  Zbl 1023.46031
[7] Haïm Brezis & Louis Nirenberg, “Degree theory and BMO, Part I : compact manifolds without boundaries”, Selecta Math. , p. 197-263  MR 1354598 |  Zbl 0852.58010
[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. 80 no. 1, p. 60-75 Article |  MR 960223 |  Zbl 0657.46027
[10] Miguel Escobedo, “Some remarks on the density of regular mappings in Sobolev classes of $S^M$-valued functions”, Rev. Mat. Univ. Complut. Madrid 1 no. 1-3, p. 127-144  MR 977045 |  Zbl 0678.46028
[11] Herbert Federer & Wendell H. Fleming, “Normal and integral currents”, Ann. of Math. (2) 72, p. 458-520  MR 123260 |  Zbl 0187.31301
[12] Emilio Gagliardo, “Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili”, Rend. Sem. Mat. Univ. Padova 27, p. 284-305 Numdam |  MR 102739 |  Zbl 0087.10902
[13] Emilio Gagliardo, “Ulteriori proprietà di alcune classi di funzioni in più variabili”, Ricerche Mat. 8, p. 24-51  MR 109295 |  Zbl 0199.44701
[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. 39 no. 2, p. 107-129  MR 2748341 |  Zbl 1207.58012
[15] Piotr Hajłasz, “Approximation of Sobolev mappings”, Nonlinear Anal. 22 no. 12, p. 1579-1591 Article |  MR 1285094 |  Zbl 0820.46028
[16] Fengbo Hang, “Density problems for $W^{1,1}(M,N)$”, Comm. Pure Appl. Math. 55 no. 7, p. 937-947 Article |  MR 1894159 |  Zbl 1020.58010
[17] Lars Inge Hedberg, “On certain convolution inequalities”, Proc. Amer. Math. Soc. 36, p. 505-510  MR 312232 |  Zbl 0283.26003
[18] Robert Hardt, David Kinderlehrer & Fang-Hua Lin, “Stable defects of minimizers of constrained variational principles”, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 no. 4, p. 297-322 Numdam |  Zbl 0657.49018
[19] Fengbo Hang & Fanghua Lin, “Topology of Sobolev mappings. II”, Acta Math. 191 no. 1, p. 55-107  MR 2020419 |  Zbl 1061.46032
[20] Fengbo Hang & Fanghua Lin, “Topology of Sobolev mappings. III”, Comm. Pure Appl. Math. 56 no. 10, p. 1383-1415 Article |  MR 1988894 |  Zbl 1038.46026
[21] Vladimir Mazʼya, Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften 342, Springer Article |  MR 2777530 |  Zbl 1217.46002
[22] Petru Mironescu, Sobolev maps on manifolds: degree, approximation, lifting, in Henri Berestycki, Michiel Bertsch, Felix E. Browder, Louis Nirenberg, Lambertus A. Peletier, Laurent Véron, éd., Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., In honor of Haïm Brezis, p. 413-436 Article |  MR 2376670 |  Zbl 1201.46032
[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. 2 no. 1, p. 113-125 Article |  MR 1890884 |  Zbl 1006.46024
[24] Domenico Mucci, “Strong density results in trace spaces of maps between manifolds”, Manuscripta Math. 128 no. 4, p. 421-441 Article |  MR 2487434 |  Zbl 1171.58002
[25] Louis Nirenberg, “On elliptic partial differential equations”, Ann. Scuola Norm. Sup. Pisa (3) 13, p. 115-162 Numdam |  MR 109940 |  Zbl 0088.07601
[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 $\pi _1(N)$”, Ann. Global Anal. Geom. 23 no. 1, p. 1-12 Article |  MR 1952855 |  Zbl 1040.58002
[28] Mohammad Reza Pakzad & Tristan Rivière, “Weak density of smooth maps for the Dirichlet energy between manifolds”, Geom. Funct. Anal. 13 no. 1, p. 223-257 Article |  Zbl 1028.58008
[29] Tristan Rivière, “Dense subsets of $H^{1/2}(S^2,S^1)$”, Ann. Global Anal. Geom. 18 no. 5, p. 517-528 Article |  MR 1790711 |  Zbl 0960.35022
[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 3, Walter de Gruyter & Co. Article |  MR 1419319 |  Zbl 0873.35001
[31] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press  MR 290095 |  Zbl 0207.13501
[32] Richard Schoen & Karen Uhlenbeck, “Boundary regularity and the Dirichlet problem for harmonic maps”, J. Differential Geom. 18 no. 2, p. 253-268  MR 710054 |  Zbl 0547.58020
[33] Brian White, “Infima of energy functionals in homotopy classes of mappings”, J. Differential Geom. 23 no. 2, p. 127-142  MR 845702 |  Zbl 0588.58017
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