Displacement Convexity Research Articles

Quantitative homogenization of the disordered $\nabla \phi $ model

We study the $\nabla \phi$ model with uniformly convex Hamiltonian $\mathcal{H} (\phi) := \sum V(\nabla \phi)$ and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the $L^2$-norm for the field subject to affine boundary condition. One of our motivations is to develop a new toolbox for studying this problem that does not rely on the Helffer-Sj\ostrand representation. Instead, we make use of the variational formulation of the partition function, the notion of displacement convexity from the theory of optimal transport, and the recently developed theory of quantitative stochastic homogenization.

Optimal density evolution with congestion: L∞ bounds via flow interchange techniques and applications to variational Mean Field Games

We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games (MFG) with a variational structure. We prove L∞ regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent’s trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L∞ results have already been found by P.-L. Lions in his course on MFG, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) φ, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics.

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.

Displacement convexity for the entropy in semi-discrete non-linear Fokker–Planck equations

The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms ofa prioriestimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

A New Class of First Order Displacement Convex Functionals

New displacement convex functionals on $S^1$ are presented, and a novel Eulerian extension of the Otto calculus is developed. The proof of the displacement convexity of these functionals relies on this Eulerian calculus, which extends the Otto calculus from a purely Riemannian setting to general Lagrangians and provides a systematic way to compute displacement Hessians of functionals involving derivatives of densities. This resolves, in the special cases of $\mathbb{R}$ and $S^1$, a problem posed by Villani in Open Problem 15.11 of [Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009].

Dealing with moment measures via entropy and optimal transport

A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ on Rd which can be expressed as the moment measures of suitable convex functions u, i.e. are of the form (∇u)#e−u for u:Rd→R∪{+∞} and finds the corresponding u by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities ρ and the optimizer ρ turns out to be e−u. This requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport. The notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers.

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM FOR A SPATIAL MODEL OF SOCIAL INTERACTIONS*

We extend Beckmann's spatial model of social interactions to the case of a two‐dimensional spatial economy with a large class of utility functions, accessing costs, and space‐dependent amenities. We show that spatial equilibria derive from a potential functional. By proving the existence of a minimizer of the functional, we obtain that of spatial equilibrium. Under mild conditions on the primitives of the economy, the functional is shown to satisfy displacement convexity. Moreover, the strict displacement convexity of the functional ensures the uniqueness of equilibrium. Also, the spatial symmetry of equilibrium is derived from that of the primitives of the economy.

Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure

We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vázquez [1,2], where the pressure is obtained as a Riesz potential associated with the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states.

On local well-posedness of the thin-film equation via the Wasserstein gradient flow

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

Uniqueness for Keller-Segel-type chemotaxis models

We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal—by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.

Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

We construct geodesics in the Wasserstein space of probability measures along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincaré inequality is implied by the weak displacement convexity of the functional.

A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval

We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott–Villani–Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.

A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm, and Villani could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.

Calculus of Variations

This workshop attracted 44 participants, some 30 of them women. Its main themes could be divided into three large groups (i) differential geometry; (ii) physics and materials; (iii) optimal transportation and its applications. The first general area encompassed the role of calculus of variations in differential geometry, including minimal surface theory and general relativity. One of the highlights was a substantial simplification of Almgren's formidable regularity theory for Q -valued functions, presented by Camillo DeLellis. This theory restricts the singularities which can occur on area minimizing currents in codimension higher than one. Another highlight was the confirmation by Del Pino (with Kowalczyk and Wei) of the counterexample long expected in dimensions N \ge 9 to DeGiorgi's conjecture about the symmetry of transition profiles in the standard model of phase segregation. Further progress showcased at the 2008 meeting included a new approach to regularity theory for harmonic maps (Rivière, Struwe), and new existence and regularity results of Willmore surfaces from a PDE point of view (Kuwert–Schätzle, Rivière). Exciting developments were also discussed concerning the structure of space-like surfaces in general relativity. Foliations by constant mean curvature surfaces and by Willmore surfaces are being used to describe the center of mass and quasi-local energy distribution of isolated gravitating systems: Tobias Lamm and Jan Metzger described their recent work with Schulze showing that Willmore surfaces behave in a similar way and can serve as an asymptotic center of mass. Turning to physics and materials science, the highlights included a review presented by Elliott Lieb of variational approaches to calculating approximate quantum ground states for atoms and molecules. A related challenge in applied mathematics is to find a satisfactory explanation for the emergence of periodic patterns in physical models, a problem which has received important recent contributions described by Andrea Malchiodi, Sylvia Serfaty, and Gero Friesecke. These involved blending techniques from the calculus of variations with other kinds of mathematics — in one case as far off as number theory. Another highlight, falling somewhere between physics and the optimization problems described below, was the idiosyncratic lecture by Eitan Bachmat explaining how supermarket queueing, airline boarding, and optimal controlling of disk drive read/write operations can be boiled down to variational problems concerning the longest future directed geodesic in a Lorentzian spacetime. Turning to the theory of optimal transportation and its connections to geometry, we recall the conference's opening lecture by Cedric Villani, addressing local to global curvature principles. In it he described his own work with John Lott (and independently Sturm), using the displacement convexity from optimal transportation to develop a theory of Ricci bounds in metric measure spaces. At the same occasion, he discussed the stability of the curvature conditions used by Ma–Trudinger–Wang to prove the regularity of optimal mappings. His first theme was reprised in the talk of Peter Topping, which used displacement convexity to give a new understanding of Perelman's monotonicity of entropy along the Ricci flow. Further talks by Figalli and Kim described new regularity results concerning free boundary problems in optimal transportation, and continuity of optimal maps for non-negatively cross-curved costs. The coordinate independence of these problems leads to phenomena which are governed by invariant structures such as curvatures, thus providing a strong point of thematic and methodological contact between this topic and the more geometric questions described in the first paragraphs above. Only a small sampling from the July 2008 Oberwolfach Workshop has been mentioned above. Lectures which fell outside these three themes included those by Gianni Dal Maso, Yury Grabovsky, Manuel Ritoré, Tatiana Toro, and Tobias Weth, and there were lively informal discussions in which many young researchers participated. Still, we hope this brief summary gives some flavor for the exciting developments presented in the workshop, and which continue to take place in the Calculus of Variations.

Ricci curvature for metric-measure spaces via optimal transport

We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having \curvature bounded below by K, with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form

Example of a displacement convex functional of first order

We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.

Size Effect of Concrete in Uniaxial Tension

In order to investigate the size effect of concrete, four sizes of rectangular prisms were tested in uniaxial tension. The cross sections of the prisms were 50x100mm, 100x100mm, 200x100mm and 400x100mm. The concrete was an ordinary one with the compressive strength of 34 MPa and the maximum aggregate size of 20mm. Notches were applied on four side faces and secondary flexure was completely eliminated during the test in order to obtain the exact nominal tensile strength. The size effect was analyzed by four factors, namely tensile strength, fracture energy, critical crack opening displacement and tension softening curves. Clear size dependence was observed in critical crack opening displacement and initial convexity of tension softening curves, and a slight size dependence was observed in tensile strength. On the other hand, size effect was unclear in fracture energy and other part of tension softening curves because of their scatters.