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Search: MSC category 49 ( Calculus of variations and optimal control; optimization )

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1. CJM Online first

Carlen, Eric; Maggi, Francesco
Stability for the Brunn-Minkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry
We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.

Keywords:Brunn-Minkowski inequality, Riesz rearrangement, Gaussian Concentration, Gates-Penrose-Lebowitz energy

2. CJM Online first

Cordero-Erausquin, Dario
Transport inequalities for log-concave measures, quantitative forms and applications
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure. We discuss quantitative forms of these inequalities, with application to the variance Brascamp-Lieb inequality.

Keywords:log-concave measures, transport inequality, Brascamp-Lieb inequality, quantitative inequalities
Categories:52A40, 60E15, 49Q20

3. CJM 2014 (vol 67 pp. 942)

Roth, Oliver
Pontryagin's Maximum Principle for the Loewner Equation in Higher Dimensions
In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin's maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$.

Keywords:univalent function, Loewner's equation
Categories:32H02, 30C55, 49K15

4. CJM 2014 (vol 67 pp. 350)

Colombo, Maria; De Pascale, Luigi; Di Marino, Simone
Multimarginal Optimal Transport Maps for One-dimensional Repulsive Costs
We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

Keywords:Monge-Kantorovich problem,optimal transport problem, cyclical monotonicity
Categories:49Q20, 49K30

5. CJM 2014 (vol 67 pp. 90)

Bousch, Thierry
Une propriété de domination convexe pour les orbites sturmiennes
Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$-periodic sequence of integers ($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter (and also $N$-periodic, consequently). It is shown that, for affine functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$-periodic sequences), the average of $|\alpha|$ on the orbit of ${\bf x}$ is greater than its average on the orbit of ${\bf s}$.

Keywords:suite sturmienne, domination convexe, optimisation ergodique
Categories:37D35, 49N20, 90C27

6. CJM 2013 (vol 65 pp. 740)

Bernard, P.; Zavidovique, M.
Regularization of Subsolutions in Discrete Weak KAM Theory
We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of $C^{1,1}$ subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.

Keywords:discrete subsolutions, regularity

7. CJM 2012 (vol 65 pp. 757)

Delanoë, Philippe; Rouvière, François
Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved
The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvature
Categories:53C35, 53C21, 53C26, 49N60

8. CJM 2011 (vol 64 pp. 924)

McCann, Robert J.; Pass, Brendan; Warren, Micah
Rectifiability of Optimal Transportation Plans
The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Categories:49K20, 49K60, 35J96, 58C07

9. CJM 2011 (vol 64 pp. 1058)

Plakhov, Alexander
Optimal Roughening of Convex Bodies
A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton's minimal resistance problem: given two bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2 \subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the resistance in the class of connected bodies $B$ such that $C_1 \subset B \subset C_2$. We prove that the infimum of resistance is zero; that is, there exist "almost perfectly streamlined" bodies.

Keywords:billiards, shape optimization, problems of minimal resistance, Newtonian aerodynamics, rough surface
Categories:37D50, 49Q10

10. CJM 2009 (vol 62 pp. 242)

Azagra, Daniel; Fry, Robb
A Second Order Smooth Variational Principle on Riemannian Manifolds
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

Keywords:smooth variational principle, Riemannian manifold
Categories:58E30, 49J52, 46T05, 47J30, 58B20

11. CJM 2009 (vol 62 pp. 320)

Jerrard, Robert L.
Some Rigidity Results Related to Monge—Ampère Functions
The space of Monge-Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2 u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampère functions. We also prove that if a Monge-Ampère function $u$ on a bounded set $\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge-Ampère function of $2$ variables.

Categories:49Q15, 53C24

12. CJM 2004 (vol 56 pp. 776)

Lim, Yongdo
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$

Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform
Categories:15A48, 49R50, 15A18, 53C3

13. CJM 2001 (vol 53 pp. 1174)

Loewen, Philip D.; Wang, Xianfu
A Generalized Variational Principle
We prove a strong variant of the Borwein-Preiss variational principle, and show that on Asplund spaces, Stegall's variational principle follows from it via a generalized Smulyan test. Applications are discussed.

Keywords:variational principle, strong minimizer, generalized Smulyan test, Asplund space, dimple point, porosity

14. CJM 1999 (vol 51 pp. 470)

Bshouty, D.; Hengartner, W.
Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations
In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta$, onto a simply connected domain $\Omega$ containing infinity and which are solutions of the system of elliptic partial differential equations $\fzbb = a(z)f_z(z)$ where the second dilatation function $a(z)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

Keywords:harmonic mappings, minimal surfaces
Categories:30C55, 30C62, 49Q05

15. CJM 1999 (vol 51 pp. 250)

Combari, C.; Poliquin, R.; Thibault, L.
Convergence of Subdifferentials of Convexly Composite Functions
In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlev\'e-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

Keywords:epi-convergence, Mosco convergence, Painlevé-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions
Categories:49A52, 58C06, 58C20, 90C30

16. CJM 1999 (vol 51 pp. 26)

Fabian, Marián; Mordukhovich, Boris S.
Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

Keywords:nonsmooth analysis, Banach spaces, separable reduction, Fréchet-like normals and subdifferentials, supporting properties, Asplund spaces
Categories:49J52, 58C20, 46B20

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