Surface energies in a two-dimensional mass-spring model for crystals

Florian Theil

Displaying similar documents to “Surface energies in a two-dimensional mass-spring model for crystals”

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...

Higher-order phase transitions with line-tension effect

Bernardo Galvão-Sousa (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [ 4 (1987) 487–512], and in a different form by Alberti in [ is a scalar density function and and are double-well potentials, the exact scaling...

Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

Luca Mugnai (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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We establish some new results about the -limit, with respect to the -topology, of two different (but related) phase-field approximations ${ℰ_{}}, {{\tilde{ℰ}}_{}}$ ℰ ε ε , x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the-limit as → 0 of ℰ, and show that in general the -limits of ℰand $\tilde{ℰ}$ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the-limit...

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)...

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...

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...

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...

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

Hang Yu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients , in three dimensions, $div (μ (\nabla u + \nabla u^{t})) + \nabla (λ div u) + V u = 0$ where and are Lipschitz continuous and . The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

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...

Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy

Jean Brossard, Christophe Leuridan (2012)

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

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Let be a measurable transformation of a probability space $(E, ℰ, π)$ , preserving the measure. Let be a random variable with law . Call (⋅, ⋅) a regular version of the conditional law of given (). Fix $B \in ℰ$ . We first prove that if is reachable from -almost every point for a Markov chain of kernel , then the -orbit of -almost every point visits . We then apply this result to the Lévy transform, which transforms the Brownian motion into the Brownian motion || − , where is the local time at 0...

Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Antoine Gloria (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the -dimensional lattice $ℤ^{d}$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector...

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

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...

A bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials

Stéphane Laurent, Catherine Legrand (2012)

ESAIM: Probability and Statistics

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In many applications, we assume that two random observations and are generated according to independent Poisson distributions $(λ S)$ x1d4ab;() and $(μ T)$ x1d4ab;() and we are interested in performing statistical inference on the ratio = / of the two incidence rates. In vaccine efficacy trials, and are typically the numbers of cases in the vaccine and the control groups respectively, is called the relative risk and the statistical model is called ‘partial immunity model’. In this paper we...

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 = ∑ ...

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

Andrew Lorent (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

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 ⊂ ℝ the functional is $I_{ϵ} (u) = \frac{1}{2} {^{\int}}_{Ω} ϵ^{-1} {|1 - {|Du|}^{2}|}^{2} + ϵ {|D^{2} u|}^{2} d z$ where belongs to the subset of functions in $W_{0}^{2, 2} (Ω)$ whose gradient (in the sense of trace) satisfies ()· = 1 where is...

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...