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

TL;DR

This paper explores Sobolev spaces relative to measures, providing new insights into tangent spaces in one dimension, and introduces a novel compactness result with applications to a gradient-penalized transport problem.

Contribution

It offers a detailed pointwise description of tangent spaces in 1D and presents a new compactness result applicable to a novel transport problem with gradient penalization.

Findings

Pointwise tangent space description in 1D Sobolev spaces

New compactness result for Sobolev spaces with measures

Application to a gradient-penalized transport problem

Abstract

We recall some known and present several new results about Sobolev spaces defined with respect to a measure, in particular a precise pointwise description of the tangent space to this measure in dimension 1. This allows to obtain an interesting, original compactness result which stays open in R^d, d \textgreater{} 1, and can be applied to a new transport problem, with gradient penalization.