Calculus of Variations and Geometric Measure Theory
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J. Louet

Some results on Sobolev spaces with respect to a measure and applications to a new transport problem

created by louet on 27 Sep 2012
modified on 25 Jan 2014


Published Paper

Inserted: 27 sep 2012
Last Updated: 25 jan 2014

Journal: Journal of Mathematical Sciences
Volume: 196
Number: 2
Pages: 152-164
Year: 2014
Doi: 10.1007/s10958-013-1647-4

Proceedings of the conference "Monge-Kantorovich optimal transportation problem, transport metrics and their applications" (EIMI, St-Petersburg, June 2012)


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.


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