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

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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},
     publisher = {mathdoc},
     volume = {411},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a3/}
}
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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/