Geodesic
Some results on Sobolev spaces with respect to a measure and applications to a new transport problem
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 63-84 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We recall some known and present several new results about Sobolev spaces defined with respect to a measure $\mu$, in particular a precise pointwise description of the tangent space to $\mu$ in dimension 1. This allows to obtain an interesting, original compactness result which stays open in $\mathbb R^d$, $d>1$, and can be applied to a new transport problem, with gradient penalization. 
@article{ZNSL_2013_411_a3,
     author = {J. Louet},
     title = {Some results on {Sobolev} spaces with respect to a~measure and applications to a~new transport problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {63--84},
     year = {2013},
     volume = {411},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a3/}
}
TY  - JOUR
AU  - J. Louet
TI  - Some results on Sobolev spaces with respect to a measure and applications to a new transport problem
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 63
EP  - 84
VL  - 411
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a3/
LA  - en
ID  - ZNSL_2013_411_a3
ER  - 
%0 Journal Article
%A J. Louet
%T Some results on Sobolev spaces with respect to a measure and applications to a new transport problem
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 63-84
%V 411
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a3/
%G en
%F ZNSL_2013_411_a3
J. Louet. Some results on Sobolev spaces with respect to a measure and applications to a new transport problem. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 63-84. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a3/

[1] S. Angenent, S. Haker, R. Kikinis, A. Tannenbaum, “On area preserving maps of minimal distorsion”, System theory: modeling, analysis and control, Kluwer Acad. Publ., 2000, 275–286 | DOI | MR | Zbl

[2] G. Bouchitté, G. Buttazzo, P. Seppecher, “Energies with respect to a measure and applications to low dimensional structures”, Calc. Var., 5 (1997), 37–54 | DOI | MR | Zbl

[3] G. Bouchitté, I. Fragalà, “Second-order energies on thin structures: variational theory and non-local effects”, J. Funct. Anal., 204:1 (2003), 228–267 | DOI | MR | Zbl

[4] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Springer-Verlag, 1977 | MR | Zbl

[5] I. Fragalà, Tangential calculus and variational integrals with respect to a measure, Ph. D. thesis, 2000 http://cvgmt.sns.it/paper/518/

[6] I. Fragalà, C. Mantegazza, “On some notions of tangent space to a measure”, Proc. Roy Soc. Edinburgh A, 129 (1999), 331–342 | DOI | MR | Zbl

[7] P. Hajłasz, “Sobolev Spaces on an Arbitrary Metric Space”, Potential Analysis, 5 (1996), 403–415 | MR | Zbl

[8] P. Hajłasz, “Sobolev spaces on metric-measure spaces”, Contemp. Math., 338 (2003), 173–218 | DOI | MR | Zbl

[9] P. Hajłasz, P. Koskela, “Sobolev met Poincaré”, Mem. Amer. Math. Soc., 688 (2000), 1–101 | MR

[10] A. Kufner, B. Opic, “How to define reasonably weighted Sobolev spaces”, Comment. Math. Univ. Carol., 25:3 (1984), 537–554 | MR | Zbl

[11] J. Louet, F. Santambrogio, “A sharp inequality for transport maps in $W^{1,p}(\mathbb R)$ via approximation”, Applied Math. Letters, 25 (2012), 648–653 | DOI | MR | Zbl

[12] D. Preiss, “Geometry of measures on $\mathbb R^n$: distribution, rectifiability and densities”, Ann. Math., 125 (1987), 573–643 | DOI | MR

[13] N. Shanmugalingam, “Newtonian spaces: an extension of Sobolev spaces to metric measure spaces”, Rev. Mat. Iberoamericana, 16 (2000), 243–279 | DOI | MR | Zbl

[14] C. Villani, Optimal transport: Old and New, Springer-Verlag, 2008 | MR

[15] V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications”, Sb. Math., 191 (2000), 973–1014 | DOI | MR | Zbl

[16] V. V. Zhikov, “Homogenization of elasticity problems on singluar structures”, Izv. Math., 66:2 (2002), 299–365 | DOI | MR | Zbl