*Published Paper*

**Inserted:** 6 oct 2011

**Last Updated:** 5 jul 2012

**Journal:** Applied Math Letters

**Volume:** 25

**Number:** 3

**Pages:** 648-653

**Year:** 2012

**Abstract:**

For $f$ convex and increasing, we prove the inequality $ \int f(

U'

) \geq \int f(nT')$, every time that $U$ is a Sobolev function of one variable and $T$ is the non-decreasing map defined on the same interval with the same image measure as $U$, and the function $n(x)$ takes into account the number of pre-images of $U$ at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.

**Keywords:**
calculus of variations, coarea formula, Semi-continuity, Monotone transport, Volume constraints

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