FIGALLI , Alessio

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity of the nonfocal...

International audience

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics,...

Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class Ck+1 in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then, there exists a potential V : M → R of class Ck−1, small in C2 topology,...

Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x ̄, and a critical viscosity subsolution u, such that u is a C1 critical solution in an open neighborhood of the positive orbit of x ̄. Suppose further that u is "C2 at x ̄". Then...

Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x ̄, and a critical viscosity subsolution u, such that u is a C1 critical solution in an open neighborhood of the positive orbit of x ̄. Suppose further that u is "C2 at x ̄". Then...

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity of the nonfocal...

Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class Ck+1 in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then, there exists a potential V : M → R of class Ck−1, small in C2 topology,...

Given a smooth compact Riemannian surface, we prove that if a suitable convexity assumption on the tangent focal cut loci is satisfied, then all injectivity domains are semiconvex.

We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.

In this paper we investigate the regularity of optimal transport maps for the squared distance cost on Riemannian manifolds. First of all, we provide some general necessary and sufficient conditions for a Riemannian manifold to satisfy the so-called Transport Continuity Property. Then, we show that on surfaces these conditions coincide....

Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani.

Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani.

We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.

Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.

In this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show that in dimension two uniqueness is far from being true. In the case of the two-dimensional disc we study solutions to Euler equations where particles...

Given a Tonelli Hamiltonian of class C2 on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the C2 topology for which the Aubry set is hyperbolic in its energy level.

In this paper we investigate the regularity of optimal transport maps for the squared distance cost on Riemannian manifolds. First of all, we provide some general necessary and sufficient conditions for a Riemannian manifold to satisfy the so-called Transport Continuity Property. Then, we show that on surfaces these conditions coincide....

International audience

Given a smooth compact Riemannian surface, we prove that if a suitable convexity assumption on the tangent focal cut loci is satisfied, then all injectivity domains are semiconvex.

We investigate the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma-Trudinger-Wang condition is stable under C 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the...

We investigate the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma-Trudinger-Wang condition is stable under C 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the...

Given a Tonelli Hamiltonian of class C2 on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the C2 topology for which the Aubry set is hyperbolic in its energy level.

In this paper we study the semiclassical limit of the Schrodinger equation. Under mild regularity assumptions on the potential U, which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an...

In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for ``almost all'' initial data, with respect to an...