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Search Results: 1 - 10 of 1026 matches for " Alessio Figalli "
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High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces
Alberto Abbondandolo,Alessio Figalli
Mathematics , 2006,
Abstract: Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic growth in the velocities or in the momenta. Such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions. In this note we show how these results can be extended to the classical setting of Tonelli Lagrangians (Lagrangians which are $C^2$-convex and superlinear in the velocities), or to Hamiltonians which are superlinear in the momenta and have a coercive action integrand.
Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index
Alberto Abbondandolo,Alessio Figalli
Mathematics , 2007,
Abstract: We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, the asymptotic Maslov indices of periodic orbits are dense in the positive half line. Furthermore, if the Hamiltonian is the Fenchel dual of an electro-magnetic Lagrangian, every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather's theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector.
Optimal transportation on non-compact manifolds
Albert Fathi,Alessio Figalli
Mathematics , 2007,
Abstract: In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type $d^r,r>1$, where $d$ is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.
Lecture notes on variational models for incompressible Euler equations
Luigi Ambrosio,Alessio Figalli
Mathematics , 2010,
Abstract: These notes briefly summarize the lectures for the Summer School "Optimal transportation: Theory and applications" held by the second author in Grenoble during the week of June 22-26, 2009. Their goal is to describe some recent results on Brenier's variational models for incompressible Euler equation.
Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces
Luigi Ambrosio,Alessio Figalli
Mathematics , 2010,
Abstract: We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is formulated in terms of the pointwise behaviour of the Ornstein-Uhlembeck semigroup.
Regularity of solutions to the parabolic fractional obstacle problem
Luis Caffarelli,Alessio Figalli
Mathematics , 2011,
Abstract: In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.
Total Variation Flow and Sign Fast Diffusion in one dimension
Matteo Bonforte,Alessio Figalli
Mathematics , 2011,
Abstract: We consider the dynamics of the Total Variation Flow (TVF) $u_t=\div(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=\Delta\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.
A sharp stability result for the relative isoperimetric inequality inside convex cones
Alessio Figalli,Emanuel Indrei
Mathematics , 2012,
Abstract: The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal C$. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal C$. Our proof follows the line of reasoning in \cite{Fi}, though several new ideas are needed in order to deal with the lack of translation invariance in our problem.
Regularity and Bernstein-type results for nonlocal minimal surfaces
Alessio Figalli,Enrico Valdinoci
Mathematics , 2013,
Abstract: We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$.
Universality in several-matrix models via approximate transport maps
Alessio Figalli,Alice Guionnet
Mathematics , 2014,
Abstract: We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices, i.e., they are given by the Sine-kernel in the bulk and the Tracy-Widom distribution at the edge. Moreover, we prove universality for the local statistics of eigenvalues of self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.
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