A new H(div)-conforming p-interpolation operator in two dimensions

Alexei Bespalov; Norbert Heuer

Displaying similar documents to “A new H(div)-conforming p-interpolation operator in two dimensions”

Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

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Let $𝔽_{}^{}$ F be a filtration andbe a random time. Let $𝔾_{}^{}$ G be the progressive enlargement of $𝔽_{}^{}$ F with. We study the following formula, called the optional splitting formula: For any $𝔾_{}^{}$ G-optional process, there exists an $𝔽_{}^{}$ F-optional process and a function defined on [0∞] × (ℝ × ) being $ℬ [0, \infty] \otimes 𝒪 (𝔽_{}^{})$ ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y = Y^{'} 1_{[0, τ)}^{} + Y^{''} (τ) 1_{[τ, \infty)}^{} .$ Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random times ...

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...

Some problems in automata theory which depend on the models of set theory

Olivier Finkel (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an -language $L (𝒜)$ (x1d49c;) accepted by a Büchi 1-counter automaton $𝒜$ x1d49c;. We prove the following surprising result: there exists a 1-counter Büchi automaton $𝒜$ x1d49c; such that the cardinality of the complement $L {(𝒜)}^{-}$ (𝒜) of the -language...

Expansions for the distribution of M-estimates with applications to the Multi-Tone problem

Christopher S. Withers, Saralees Nadarajah (2011)

ESAIM: Probability and Statistics

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We give a stochastic expansion for estimates $\hat{θ}$ that minimise the arithmetic mean of (typically independent) random functions of a known parameter. Examples include least squares estimates, maximum likelihood estimates and more generally -estimates. This is used to obtain leading cumulant coefficients of $\hat{θ}$ needed for the Edgeworth expansions for the distribution and density ) to magnitude (or to for the symmetric...

Exact null internal controllability for the heat equation on unbounded convex domains

Viorel Barbu (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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The liner parabolic equation $\frac{y}{t} - \frac{1}{2} 𝔻 y + F \cdot y = {\vec{1}}_{_{0}} u$ ∂y ∂t − 1 2 Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝ with smooth boundary is exactly null controllable on each finite interval if 𝒪is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of x1d4aa; . Here, : ℝ → ℝ is a bounded, -continuous function, and = ∇, where is convex and coercive.

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.

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...

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...

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...

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)...

Coarse quantization for random interleaved sampling of bandlimited signals

Alexander M. Powell, Jared Tanner, Yang Wang, Özgür Yılmaz (2012)

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

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The compatibility of unsynchronized interleaved uniform sampling with Sigma-Delta analog-to-digital conversion is investigated. Let be a bandlimited signal that is sampled on a collection of interleaved grids { + } with offsets ${T_{n}}_{n = 1}^{N} \subset [0, T]$ T n n = 1 N ⊂ [ 0 ,T ] . If the offsets are chosen independently and uniformly at random from [0] and if the sample values of are quantized with a first order Sigma-Delta algorithm, then with high probability the quantization...

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...

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Benjamin Charlier (2013)

ESAIM: Probability and Statistics

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Let ( $𝕊^{1}, d_{𝕊^{1}}$ S 1 , d S 1 ) be the unit circle in ℝ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure to admit a well defined Fréchet mean on ( $𝕊^{1}, d_{𝕊^{1}}$ S 1 , d S 1 ). We derive a new sufficient condition of existence() with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space (ℝ)

T. Suslina (2010)

Mathematical Modelling of Natural Phenomena

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In (ℝ; ℂ), we consider a wide class of matrix elliptic second order differential operators $𝒜$ with rapidly oscillating coefficients (depending on /). For a fixed > 0 and small > 0, we find approximation of the operator exponential exp(− $𝒜$ ) in the ( (ℝ; ℂ) → (ℝ; ℂ))-operator norm...

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...

Curve cuspless reconstruction via sub-riemannian geometry

Ugo Boscain, Remco Duits, Francesco Rossi, Yuri Sachkov (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the problem of minimizing $\int_{0}^{ℓ} \sqrt{ξ^{2} + K^{2} (s)} d s$ ∫ 0 ℓ ξ 2 + K 2 ( s ) d s for a planar curve having fixed initial and final positions and directions. The total length is free. Here is the arclength parameter, () is the curvature of the curve and > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions...

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...

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

Florian Theil (2011)

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

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We study an atomistic pair potential-energy () that describes the elastic behavior of two-dimensional crystals with atoms where $y \in ℝ^{2 \times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of admits an asymptotic expansion involving fractional powers of : $\min_{y} E^{(n)} (y) = n E_{bulk} + \sqrt{n} E_{surface} + o (\sqrt{n}), n \to \infty .$ The bulk energy density ...

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...

Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift

Pierre Étoré, Miguel Martinez (2014)

ESAIM: Probability and Statistics

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In this note we propose an exact simulation algorithm for the solution of (1) $d X_{t} = d W_{t} + \bar{b} (X_{t}) d t, X_{0} = x,$ d X t = d W t + b̅ ( X t ) d t, X 0 = x, where $\bar{b}$ b̅is a smooth real function except at point 0 where $\bar{b} (0 +) \neq \bar{b} (0 -)$ b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew...

A new kind of augmentation of filtrations

Joseph Najnudel, Ashkan Nikeghbali (2011)

ESAIM: Probability and Statistics

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Let (Ω, $ℱ$ , ( $ℱ_{t}$ ), $ℙ$ ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $ℱ_{\infty}$ (the-algebra generated by ( $ℱ_{t}$ )) a coherent family of probability measures ( $ℚ_{t}$ ) indexed by , each of them being defined on $ℱ_{t}$ . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative...

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