In this talk I will describe a variant of the classical optimal transport problem where, given two densities of mass f and g, one considers the problem of transporting a fixed fraction of the mass of f onto g as cheaply as possible. First, I will show some existence and uniqueness results. Then we will see that this problem corresponds to a Monge-Ampere equation in a domain with free boundary, and we will study the regularity property of the optimal transport map and the free boundary.
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