Expand all Collapse all | Results 1 - 14 of 14 |
1. CJM Online first
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 |
2. CJM Online first
Multimarginal Optimal Transport Maps for $1$-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 |
3. CJM Online first
Une propriÃ©tÃ© 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 |
4. CJM 2013 (vol 65 pp. 740)
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 Category:49C15 |
5. CJM 2012 (vol 65 pp. 757)
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 |
6. CJM 2011 (vol 64 pp. 1058)
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 |
7. CJM 2011 (vol 64 pp. 924)
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 |
8. CJM 2009 (vol 62 pp. 242)
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 |
9. CJM 2009 (vol 62 pp. 320)
Some Rigidity Results Related to MongeâAmpÃ¨re Functions The space of Monge-AmpÃ¨re 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-AmpÃ¨re functions. We also
prove that if a Monge-AmpÃ¨re 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-AmpÃ¨re function of $2$ variables.
Categories:49Q15, 53C24 |
10. CJM 2004 (vol 56 pp. 776)
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 |
11. CJM 2001 (vol 53 pp. 1174)
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 Category:49J52 |
12. CJM 1999 (vol 51 pp. 470)
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 |
13. CJM 1999 (vol 51 pp. 250)
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, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 |
14. CJM 1999 (vol 51 pp. 26)
Separable Reduction and Supporting Properties of FrÃ©chet-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, FrÃ©chet-like normals and subdifferentials, supporting properties, Asplund spaces Categories:49J52, 58C20, 46B20 |