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47
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 34 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
Existence and uniqueness for dislocation dynamics with nonnegative velocity
, 2004
"... We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we ..."
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Cited by 25 (15 self)
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We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows to prove that the energy associated to the dislocation dynamics is non increasing. AMS Classification: 35F25, 35D05. Keywords: Dislocation dynamics, eikonal equation, HamiltonJacobi equations,
Optimal transportation on noncompact manifolds
"... In this work, we show how to obtain for noncompact 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 dr, r> 1, where d is the Riemannian distan ..."
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Cited by 22 (6 self)
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In this work, we show how to obtain for noncompact 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 dr, r> 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction. 1
Nearly round spheres look convex
 of Progress in Mathematics
"... Abstract. 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 t ..."
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Cited by 13 (4 self)
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Abstract. 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.
Regularity and variationality of solutions to HamiltonJacobi equations. part i: regularity
 ESAIM Control Optim. Calc. Var
"... We formulate an Hamilton–Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of convexity of the sets {p: H(x, p) ≤ 0} ⊂ T ∗ x M, for all x. In this paper we reduce the above problem to a simpler problem: this shows that u may be built using an asy ..."
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Cited by 9 (3 self)
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We formulate an Hamilton–Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of convexity of the sets {p: H(x, p) ≤ 0} ⊂ T ∗ x M, for all x. In this paper we reduce the above problem to a simpler problem: this shows that u may be built using an asymmetric distance (this is a generalization of the “distance function ” in Finsler Geometry): this brings forth a ’completeness ’ condition, and a Hopf–Rinow theorem adapted to Hamilton–Jacobi problems. The ’completeness’ condition implies that u is the unique viscosity solution to the above problem. When H is moreover of class C 1,1, we show how the completeness condition is equivalent to a condition expressed using the characteristics equations. 7
HAMILTON–JACOBI SEMIGROUP ON LENGTH SPACES AND APPLICATIONS
, 2006
"... Abstract. We define a Hamilton–Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1 ..."
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Cited by 8 (0 self)
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Abstract. We define a Hamilton–Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length space implies a global Poincaré inequality and (2) if the space satisfies a doubling condition, a local Poincaré inequality and a log Sobolev inequality then it also satisfies a Talagrand inequality. Links between concentration of measure, log Sobolev inequalities, Talagrand inequalities and Poincaré inequalities have been studied in the setting of Riemannian manifolds [1, 2, 3, 8, 9, 12]. The main result in the paper of Otto and Villani [12] can be informally stated as follows: on a Riemannian manifold, a log Sobolev inequality implies a Talagrand inequality, which in turn implies a Poincaré (or spectral gap) inequality, all of this being without any degradation of the constants. On the other hand, there has been intense recent activity to develop a theory of Ricci curvature bounds, log Sobolev inequalities and related inequalities in the more general
On the MaTrudingerWang Curvature on Surfaces
"... 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 positi ..."
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Cited by 8 (1 self)
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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 Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.
The distance function from the boundary in a Minkowski space
 TRANS. AMER. MATH. SOC
, 2005
"... Let the space R n be endowed with a Minkowski structure M (that is M: R n → [0, +∞) is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class C 2), and let d M (x, y) be the (asymmetric) distance associated to M. Given an open domain Ω ⊂ R n o ..."
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Cited by 6 (3 self)
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Let the space R n be endowed with a Minkowski structure M (that is M: R n → [0, +∞) is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class C 2), and let d M (x, y) be the (asymmetric) distance associated to M. Given an open domain Ω ⊂ R n of class C 2, let dΩ(x): = inf{d M (x, y); y ∈ ∂Ω} be the Minkowski distance of a point x ∈ Ω from the boundary of Ω. We prove that a suitable extension of dΩ to R n (which plays the röle of a signed Minkowski distance to ∂Ω) is of class C 2 in a tubular neighborhood of ∂Ω, and that dΩ is of class C 2 outside the cut locus of ∂Ω (that is the closure of the set of points of non–differentiability of dΩ in Ω). In addition, we prove that the cut locus of ∂Ω has Lebesgue measure zero, and that Ω can be decomposed, up to this set of vanishing measure, into geodesics starting from ∂Ω and going into Ω along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point x ∈ Ω outside the cut locus the pair (p(x), dΩ(x)), where p(x) denotes the (unique) projection of x on ∂Ω, and we apply these techniques to the analysis of PDEs of MongeKantorovich type arising from problems in optimal transportation theory and shape optimization.
Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity.
"... Abstract The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to ominimal structures (Kurdyka) have a considerable impact on the analysis of gradientlike methods and related problems: minimization methods, complexity theory, asymptotic anal ..."
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Cited by 5 (3 self)
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Abstract The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to ominimal structures (Kurdyka) have a considerable impact on the analysis of gradientlike methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Lojasiewicz inequality (hereby called the KurdykaLojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by −∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the KurdykaLojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines —a concept linked to the location of the less steepest points at the level sets of f— and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the KurdykaLojasiewicz inequality. Key words Lojasiewicz inequality, gradient inequalities, metric regularity, subgradient curve, talweg, gradient method, convex functions, global convergence, proximal method.