On convex sets that minimize the average distance

Antoine Lemenant; Edoardo Mainini

Displaying similar documents to “On convex sets that minimize the average distance”

A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Andrew Lorent (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain ⊂ ℝthe functional is $I_{} (u) = \frac{1}{2} \int_{Ω}^{- 1} {| 1 - | D u |}^{2} |^{2} + {| D^{2} u |}^{2} d z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z wherebelongs to the subset of functions in $W_{0}^{2, 2} (Ω)$ W02,2(Ω) whose gradient (in the sense of trace) satisfies()· = 1 where is the inward pointing unit normal to at . In [1...

Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation

Nastasiya F. Grinberg (2013)

ESAIM: Probability and Statistics

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In this note we prove that the local martingale part of a convex function of a -dimensional semimartingale = + can be written in terms of an Itô stochastic integral ∫()d, where () is some particular measurable choice of subgradient ∇ f ( x ) of at , and is the martingale part of . This result was first proved by Bouleau in [N. Bouleau, 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for $\tilde{X} = X + ϵ B$ x10ff65; X = X + ϵB , > 0, where is...

Variational approximation of a functional of Mumford–Shah type in codimension higher than one

Francesco Ghiraldin (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we consider a new kind of Mumford–Shah functional () for maps : ℝ → ℝ with ≥ . The most important novelty is that the energy features a singular set of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy () −convergence, in the same spirit of the work by Ambrosio and Tortorelli [L....

Limit theorems for measure-valued processes of the level-exceedance type

Andriy Yurachkivsky (2011)

ESAIM: Probability and Statistics

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Let, for each ∈ , (, ۔) be a random measure on the Borel -algebra in ℝ such that E(, ℝ) < ∞ for all and let $\hat{ψ}$ (, ۔) be its characteristic function. We call the function $\hat{ψ}$ ( ,…, ; ,…, ) = $𝖤 \prod_{j = 1}^{l} \hat{ψ} (t_{j}, z_{j})$ of arguments ∈ ℕ, , … ∈ , , ∈ ℝ the of the measure-valued random function (MVRF) (۔, ۔). A general limit theorem for MVRF's in terms of covaristics is proved and...

Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

Hedy Attouch, Paul-Émile Maingé (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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In the setting of a real Hilbert space $ℋ$ , we investigate the asymptotic behavior, as time goes to infinity, of trajectories of second-order evolution equations () + $\dot{u}$ () + (()) + (()) = 0, where is the gradient operator of a convex differentiable potential function : $ℋ \to ℝ$ ,: $ℋ \to ℋ$ is a maximal monotone operator which is assumed to be-cocoercive, and > 0 is a damping parameter. Potential and non-potential effects are associated...

Relaxation in BV of integrals with superlinear growth

Parth Soneji (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We study properties of the functional $\begin{matrix} ℱ_{loc} (u, Ω) : = inf_{(u_{j})} \{\underset{j \to \infty}{lim inf} \int_{Ω} f (\nabla u_{j}) x |\begin{matrix} (u_{j}) \subset W_{loc}^{1, r} (Ω,) \\ u_{j} u in (Ω,) \end{matrix}\}, \end{matrix}$ F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;R^N), and f:R^{N × n} → R is continuous and satisfies 0 ≤ f(ξ)...

Continuity of solutions of a nonlinear elliptic equation

Pierre Bousquet (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider a nonlinear elliptic equation of the form div [(∇)] + [] = 0 on a domain Ω, subject to a Dirichlet boundary condition tr = . We do not assume that the higher order term satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and satisfies a one-sided bounded slope condition, or when is radial: $a (ξ) = l (| ξ |) | ξ | ξ$ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasing:ℝ → ℝ

The H–1-norm of tubular neighbourhoods of curves

Yves van Gennip, Mark A. Peletier (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the -norm of the function 1 on tubular neighbourhoods of curves in $ℝ^{2}$ . We take the limit of small thickness, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit → 0, containing contributions from the length of the curve (at order ), the ends ( ), and the curvature ( ). The second result is a Γ-convergence result, in which the central curve may vary along...

Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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We study integral functionals constrained to divergence-free vector fields in on a thin domain, under standard -growth and coercivity assumptions, 1 ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject...

Hydrodynamic limit of a d-dimensional exclusion process with conductances

Fábio Júlio Valentim (2012)

Annales de l'I.H.P. Probabilités et statistiques

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Fix a polynomial of the form () = + ∑2≤≤ =1 with (1) gt; 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on $𝕋^{d}$ , with conductances given by special class of functions, is described by the unique weak solution of the non-linear parabolic partial differential equation = ∑ ...

Inequality-sum: a global constraint capturing the objective function

Jean-Charles Régin, Michel Rueher (2010)

RAIRO - Operations Research

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This paper introduces a new method to prune the domains of the variables in constrained optimization problems where the objective function is defined by a sum , and where the integer variables are subject to difference constraints of the form . An important application area where such problems occur is deterministic scheduling with the as optimality criteria. This new constraint is also more general than a sum constraint defined on a set of ordered variables. Classical...

Minimising convex combinations of low eigenvalues

Mette Iversen, Dario Mazzoleni (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the variational problem inf{ () + () + (1 − − ) () | Ω open in ℝ, || ≤ 1}, for ∈ [0, 1], + ≤ 1, where () is the th eigenvalue of the Dirichlet Laplacian acting in () and || is the Lebesgue measure of . We investigate for which values of every minimiser is connected.

Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)

Marcus Wagner (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands (, , ) in presence of a convex control restriction. The relaxed problem, wherein the integrand has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image...

Means in complete manifolds: uniqueness and approximation

Marc Arnaudon, Laurent Miclo (2014)

ESAIM: Probability and Statistics

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Let be a complete Riemannian manifold, ∈ ℕ and ≥ 1. We prove that almost everywhere on = ( ,, ) ∈ for Lebesgue measure in , the measure $μ (x) = N \sum_{k = 1}^{N} x_{k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a unique–mean (). As a consequence, if = ( ,, ) is a -valued random variable with absolutely continuous law, then almost surely (()) has a unique –mean. In particular if ( ...

Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations

Luís Balsa Bicho, António Ornelas (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove of vector minimizers () = (||) to multiple integrals ∫ ((), |()|) on a ⊂ ℝ, among the Sobolev functions (·) in + (, ℝ), using a : ℝ×ℝ → [0,∞] with (·) and . Besides such basic hypotheses, (·,·) is assumed to satisfy also...

Regularization of an unilateral obstacle problem

Ahmed Addou, E. Bekkaye Mermri, Jamal Zahi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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The aim of this article is to give a regularization method for an unilateral obstacle problem with obstacle and second member , which generalizes the one established by the authors of [4] in case of null obstacle and a second member is equal to constant .

Penalization versus Goldenshluger − Lepski strategies in warped bases regression

Gaëlle Chagny (2013)

ESAIM: Probability and Statistics

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This paper deals with the problem of estimating a regression function , in a random design framework. We build and study two adaptive estimators based on model selection, applied with warped bases. We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases. Instead of expanding directly the target function on these bases, we rather consider the expansion of = ∘ , where is the cumulative distribution function of the design, following...

Exponential deficiency of convolutions of densities

Iosif Pinelis (2012)

ESAIM: Probability and Statistics

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If a probability density (x) (x ∈ ℝ) is bounded and := ∫e (x)dx < ∞ for some linear functional u and all ∈ (01), then, for each ∈ (01) and all large enough , the -fold convolution of the -tilted density ${\tilde{p}}_{t}$ ˜pt := e (x)/ is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...

Regularization of linear least squares problems by total bounded variation

G. Chavent, K. Kunisch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the problem : Minimize $λ_{2}$ over , where is a closed convex subset of (Ω), and the last additive term denotes the BV-seminorm of is a linear operator from ∩ into the observation space . We formulate necessary optimality conditions for (). Then we show that () admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized...