Optimal transport between mutually singular measures – Giuseppe Buttazzo (Universita’ di Pisa)


Sala Seminari (Dip. Matematica).


We study the Wasserstein distance between two measures $\mu,\ u$ which are mutually singular. In particular, we are interested in minimization problems of the form $$W(\mu,{\cal A})=\inf\big\{W(\mu,\ u)\ :\ \ u\in{\cal A}\big\}$$ where $\mu$ is a given probability and ${\cal A}$ is contained in the class $\mu^\perp$ of probabilities that are singular with respect to $\mu$. Several cases for ${\cal A}$ are considered; in particular, when ${\cal A}$ consists of $L^1$ densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem $$\min\big\{P(B)+kW(A,B)\ :\A\cap B=0,\A=B=1\big\},$$ where $k>0$ is a fixed constant, $P(A)$ is the perimeter of $A$, and both sets $A,B$ may vary.

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