Given compactly supported 0 f,g 2 L 1 (R n ), the problem of trans- porting a fraction m min{kfkL1,kgkL1} of the mass of f onto g as cheaply as possible is considered, where cost per unit mass transported is given by a cost function c, typically quadratic c(x,y) = |x y| 2 /2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampere equation, for which sucient conditions are given to guarantee uniqueness of the solution, such as f vanishing on sptg in the quadratic case. The part of f to be transported increases monotonically with m, and if sptf and sptg are separated by a hyperplane H, then this part will separated from the balance of f by a semiconcave Lipschitz graph over the hyperplane. If f = f and g = g are bounded away from zero and infinity on separated strictly convex domains , R n , for the quadratic cost this graph is shown to be a C 1, loc hypersurface in whose normal coincides with the direction transported; the optimal map between f and g is shown to be Holder continuous up to this free bound- ary, and to those parts of the fixed boundary @ which map to locally convex parts of the path-connected target region.

https://www.math.toronto.edu/mccann/papers/annals2010.pdf

Free boundaries in optimal transport and Monge-Ampere obstacle problems

We consider the non-nonlinear optimal transportation problem of minimizing the cost functional $\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|$ in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

The $\infty$-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps

Starting from the work by Brenier [Extended Monge-Kantorovich theory, in Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin (2003), pp. 91-121], where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kinds of cost functions are allowed. This seems relevant in some problems presenting congestion effects as, for instance, traffic on a highway, crowds moving in domains with obstacles, and, in general, in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in Benamou and Brenier [A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393].

/pdf/an-optimization-problem-for-mass-transportation-with-2ep59ea7vq.pdf

An Optimization Problem for Mass Transportation with Congested Dynamics

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella

/pdf/sharp-isoperimetric-inequalities-via-the-abp-method-11xr5803mm.pdf

Sharp isoperimetric inequalities via the ABP method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. 
We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. 
As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.

https://cvgmt.sns.it/media/doc/paper/554/FuscoMaggiPratelliMixedGauss.pdf

On the isoperimetric problem with respect to a mixed Euclidean–Gaussian density

In this paper we study the problem of finding an optimal pricing policy for the use of the public transportation network in a given populated area. The transportation network, modeled by a Borel set $\Sigma\subset \R^n$ of finite length, the densities of the population and of the services (or workplaces), modeled by the respective finite Borel measures $\varphi_0$ and $\varphi_1$, and the effective cost $A(t)$ for a citizen to cover a distance $t$ without the use of the transportation network are assumed to be given. The pricing policy to be found is then a cost $B(t)$ to cover a distance $t$ with the use of the transportation network (i.e., the "price of the ticket for a distance $t$"), and it has to provide an equilibrium between the needs of the population (hence minimizing the total cost of transportation of the population to the services/workplaces) and that of the owner of the transportation network (hence maximizing the total income of the latter). We present a model for such a choice and discuss the existence as well as some qualitative properties of the resulting optimal pricing policies.

Optimal Pricing Policies for Public Transportation Networks

We consider the problem of minimising or maximising the quantity $\lambda(O)T^q(O)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $\lambda(O)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(O)$, while $T(O)$ is the torsional rigidity of $O$. The optimisation problem above is considered in the class of {\it all domains} $O$, in the class of {\it convex domains} $O$, and in the class of {\it thin domains}. The full Blaschke-Santalo diagram for $\lambda(O)$ and $T(O)$ is obtained in dimension one, while for higher dimensions we provide some bounds.

On the relations between principal eigenvalue and torsional rigidity

We consider the problem of minimizing or maximizing the quantity λ(O)Tq(O) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, λ(O) denotes the first eigenvalue of the Di...

On relations between principal eigenvalue and torsional rigidity

Abstract We prove the validity of the ε - ε β {\\varepsilon-\\varepsilon^{\\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

Aldo Pratelli

Papers

On the isoperimetric problem with respect to a mixed Euclidean–Gaussian density

Optimal Pricing Policies for Public Transportation Networks

On the relations between principal eigenvalue and torsional rigidity

On relations between principal eigenvalue and torsional rigidity

The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets