Regularity of optimal transport maps on multiple products of spheres

Alessio Figalli; Young-Heon Kim; Robert McCann

Displaying similar documents to “Regularity of optimal transport maps on multiple products of spheres”

Synthesis of optimal control for nonlinear third order systems

Wiesław Szwiec

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The paper is devoted to a certain problem in the theory of synthesis of optimal control. The fundamental results of research in the synthesis of optimal control for second order systems are given in [1], [2], [5], [10]. In particular, in [2] and [5] some specific nonlinear systems are investigated, namely the systems related with the differential equations of the forms̈ + F(s,ṡ) = u or s̈ + F(s,ṡ,u) = 0where u is a control parameter in the interval [—1,1]. Some results concerning third...

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

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We construct biharmonic non-harmonic maps between Riemannian manifolds $(M, g)$ and $(N, h)$ by first making the ansatz that $ϕ : (M, g) \to (N, h)$ be a harmonic map and then deforming the metric on $N$ by ${\tilde{h}}_{α} = α h + (1 - α) d f \otimes d f$ to render $ϕ$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N, h)$ and $α \in (0, 1)$ . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Limiting average cost control problems in a class of discrete-time stochastic systems

Nadine Hilgert, Onesimo Hernández-Lerma (2001)

Applicationes Mathematicae

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We consider a class of $ℝ^{d}$ -valued stochastic control systems, with possibly unbounded costs. The systems evolve according to a discrete-time equation $x_{t + 1} = G ₙ (x_{t}, a_{t}) + ξ_{t}$ (t = 0,1,... ), for each fixed n = 0,1,..., where the $ξ_{t}$ are i.i.d. random vectors, and the Gₙ are given functions converging pointwise to some function $G_{\infty}$ as n → ∞. Under suitable hypotheses, our main results state the existence of stationary control policies that are expected average cost (EAC) optimal and sample path average cost (SPAC)...

On $G$ -sets and isospectrality

Ori Parzanchevski (2013)

Annales de l’institut Fourier

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We study finite $G$ -sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If $M$ is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then $M$ has isospectral non-isometric covers.

Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian (2010)

Studia Mathematica

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Recently it was proved for 1 < p < ∞ that $ω^{m} {(f, t)}_{p}$ , a modulus of smoothness on the unit sphere, and $K ̃ ₘ {(f, t^{m})}_{p}$ , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence $ω^{m} {(f, t)}_{p} \approx K ̃ ₘ {(f, t^{r})}_{p}$ does not hold either for p = ∞ or for p = 1.

Collapse of warped submersions

Szymon M. Walczak (2006)

Annales Polonici Mathematici

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We generalize the concept of warped manifold to Riemannian submersions π: M → B between two compact Riemannian manifolds $(M, g_{M})$ and $(B, g_{B})$ in the following way. If f: B → (0,∞) is a smooth function on B which is extended to a function f̂ = f ∘ π constant along the fibres of π then we define a new metric $g_{f}$ on M by $g_{f} |_{\times} \equiv g_{M} |_{\times}, g_{f} {|_{\times T M ̂} \equiv f ̂ ² g_{M} |}_{\times T M ̂}$ , where and denote the bundles of horizontal and vertical vectors. The manifold $(M, g_{f})$ obtained that way is called a warped submersion. The function f is called a warping function. We show...

Stability estimating in optimal stopping problem

Elena Zaitseva (2008)

Kybernetika

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We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $X$ . It is supposed that an unknown transition probability $p (\cdot | x)$ , $x \in X$ , is approximated by the transition probability $\tilde{p} (\cdot | x)$ , $x \in X$ , and the stopping rule ${\tilde{τ}}_{*}$ , optimal for $\tilde{p}$ , is applied to the process governed by $p$ . We found an upper bound for the difference between the total expected cost, resulting when applying ${\tilde{τ}}_{*}$ , and the minimal total expected cost. The bound given is a constant times ${sup}_{x \in X} ∥ p (\cdot | x) - \tilde{p} (\cdot | x) ∥$ , where $∥ \cdot ∥$ is the...

Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2016)

Journal of the European Mathematical Society

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We consider the wave and Schrödinger equations on a bounded open connected subset $Ω$ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $ω$ of $Ω$ during a time interval $[0, T]$ with $T > 0$ . It is well known that, if the pair $(ω, T)$ satisfies the Geometric Control Condition ( $ω$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions...

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman, Luboš Pick (2011)

Studia Mathematica

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We study imbeddings of the Sobolev space $W^{m, ϱ} (Ω)$ : = u: Ω → ℝ with $ϱ (\partial^{α} u / \partial x^{α})$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_{ϱ}$ and $σ_{ϱ}$ , that are optimal with respect to the inclusions $W^{m, ϱ} (Ω) \subset W^{m, τ_{ϱ}} (Ω) \subset L_{σ_{ϱ}} (Ω)$ . General formulas for $τ_{ϱ}$ and $σ_{ϱ}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov (2014)

Journal de l’École polytechnique — Mathématiques

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We consider the Schrödinger operator $- Δ + V (x)$ on $H_{0}^{1} (Ω)$ , where $Ω$ is a given domain of $ℝ^{d}$ . Our goal is to study some optimization problems where an optimal potential $V \geq 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Representation of equilibrium solutions to the table problem of growing sandpiles

Piermarco Cannarsa, Pierre Cardaliaguet (2004)

Journal of the European Mathematical Society

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In the dynamical theory of granular matter the so-called table problem consists in studying the evolution of a heap of matter poured continuously onto a bounded domain $Ω \subset ℝ^{2}$ . The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge–Kantorovich problem, is the object of this paper....

On the optimal setting of the $h p$ -version of the finite element method

Chleboun, Jan

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The goal of this contribution is to find the optimal finite element space for solving a particular boundary value problem in one spatial dimension. In other words, the optimal use of available degrees of freedom is sought after. This is done through optimizing both the mesh and the polynomial degree of the basis functions. The resulting combinatorial optimization problem is solved in parallel by a Matlab program running on a cluster of multi-core personal computers.

$f$ -biminimal maps between Riemannian manifolds

Yan Zhao, Ximin Liu (2019)

Czechoslovak Mathematical Journal

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We give the definition of $f$ -biminimal submanifolds and derive the equation for $f$ -biminimal submanifolds. As an application, we give some examples of $f$ -biminimal manifolds. Finally, we consider $f$ -minimal hypersurfaces in the product space $ℝ^{n} \times 𝕊^{1} (a)$ and derive two rigidity theorems.

On $2p$ -dimensional Riemannian manifolds with positive scalar curvature

Domenico Perrone (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In questo lavoro si danno alcuni risultati sugli spettri degli operatori di Laplace per varietà Riemanniane compatte con curvatura scalare positiva e di dimensione $2p$ . Ad essi si aggiunge una osservazione riguardante la congettura di Yamabe.