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Cristinel Mardare - One of the best experts on this subject based on the ideXlab platform.
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An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors
Comptes Rendus Mathematique, 2004Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let Ω be a bounded open Connected Subset of Rn with a Lipschitz-continuous boundary and let Θ∈C1(Ω;Rn) be a deformation of the set Ω satisfying det∇Θ>0 in Ω. It is established that there exists a constant C(Θ) with the following property: for each deformation Φ∈H1(Ω;Rn) satisfying det∇Φ>0 a.e. in Ω, there exist an n×n rotation matrix R=R(Φ,Θ) and a vector b=b(Φ,Θ) in Rn such that Φ−(b+RΘ)H1(Ω)⩽C(Θ)∇ΦT∇Φ−∇ΘT∇ΘL1(Ω)1/2. The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Muller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
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On rigid and infinitesimal rigid displacements in shell theory
Journal de Mathématiques Pures et Appliquées, 2004Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let ω be an open Connected Subset of R^2 and let θ be an immersion from ω into R^3. It is first established that the set formed by all rigid displacements, i.e., that preserve the metric and the curvature, of the surface θ(ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(ω). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the same surface θ(ω) is nothing but the tangent space at the origin to this submanifold. In this fashion, the “infinitesimal rigid displacement lemma on a surface”, which plays a key role in shell theory, is put in its proper perspective.
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An estimate of the $H^1$-norm of deformations in terms of the $L^1$-norm of their Cauchy-Green tensors
C. R. Math.Acad.Sci., 2004Co-Authors: Cristinel Mardare, Philippe G. CiarletAbstract:Let Ω be a bounded open Connected Subset of Rn with a Lipschitz-continuous boundary and let Θ ∈ C^1(Ω;R^n) be a deformation of the set Ω satisfying det∇Θ > 0 in Ω. It is established that there exists a constant C(Θ) with the following property: For each deformation Φ ∈ H^1(Ω; R^n) satisfying det ∇Φ > 0 a.e. in Ω, there exist an n×n rotation matrix R=R(Φ,Θ) and a vector b=b(Φ,Θ) in R^n such that ||Φ−(b+RΘ)||_{H^1(Ω)}≤ C(Θ) ||∇Φ^T ∇Φ − ∇Θ^T ∇Θ||_{L^1(Ω)}^{1/2}. The proof relies in particular on a fundamental “geometric rigidity lemma”, recently proved by G. Friesecke, R.D. James, and S. Muller.
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On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory
Comptes Rendus Mathématique, 2003Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let ω be an open Connected Subset of R^2 and let θ be an immersion from ω into R^3. It is established that the set formed by all rigid displacements of the surface θ(ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(ω). It is shown that the infinitesimal rigid displacements of the same surface θ(ω) span the tangent space at the origin to this submanifold.
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On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional elasticity
Comptes Rendus Mathématique, 2003Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let Ω be an open Connected Subset of R^3 and let Θ be an immersion from Ω into R^3. It is established that the set formed by all rigid displacements of the open set Θ(Ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(Ω). It is also shown that the infinitesimal rigid displacements of the same set Θ(Ω) span the tangent space at the origin to this submanifold.
Philippe G. Ciarlet - One of the best experts on this subject based on the ideXlab platform.
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An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors
Comptes Rendus Mathematique, 2004Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let Ω be a bounded open Connected Subset of Rn with a Lipschitz-continuous boundary and let Θ∈C1(Ω;Rn) be a deformation of the set Ω satisfying det∇Θ>0 in Ω. It is established that there exists a constant C(Θ) with the following property: for each deformation Φ∈H1(Ω;Rn) satisfying det∇Φ>0 a.e. in Ω, there exist an n×n rotation matrix R=R(Φ,Θ) and a vector b=b(Φ,Θ) in Rn such that Φ−(b+RΘ)H1(Ω)⩽C(Θ)∇ΦT∇Φ−∇ΘT∇ΘL1(Ω)1/2. The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Muller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
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On rigid and infinitesimal rigid displacements in shell theory
Journal de Mathématiques Pures et Appliquées, 2004Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let ω be an open Connected Subset of R^2 and let θ be an immersion from ω into R^3. It is first established that the set formed by all rigid displacements, i.e., that preserve the metric and the curvature, of the surface θ(ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(ω). It is then shown that the vector space formed by all the infinitesimal rigid displacements of the same surface θ(ω) is nothing but the tangent space at the origin to this submanifold. In this fashion, the “infinitesimal rigid displacement lemma on a surface”, which plays a key role in shell theory, is put in its proper perspective.
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An estimate of the $H^1$-norm of deformations in terms of the $L^1$-norm of their Cauchy-Green tensors
C. R. Math.Acad.Sci., 2004Co-Authors: Cristinel Mardare, Philippe G. CiarletAbstract:Let Ω be a bounded open Connected Subset of Rn with a Lipschitz-continuous boundary and let Θ ∈ C^1(Ω;R^n) be a deformation of the set Ω satisfying det∇Θ > 0 in Ω. It is established that there exists a constant C(Θ) with the following property: For each deformation Φ ∈ H^1(Ω; R^n) satisfying det ∇Φ > 0 a.e. in Ω, there exist an n×n rotation matrix R=R(Φ,Θ) and a vector b=b(Φ,Θ) in R^n such that ||Φ−(b+RΘ)||_{H^1(Ω)}≤ C(Θ) ||∇Φ^T ∇Φ − ∇Θ^T ∇Θ||_{L^1(Ω)}^{1/2}. The proof relies in particular on a fundamental “geometric rigidity lemma”, recently proved by G. Friesecke, R.D. James, and S. Muller.
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On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory
Comptes Rendus Mathématique, 2003Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let ω be an open Connected Subset of R^2 and let θ be an immersion from ω into R^3. It is established that the set formed by all rigid displacements of the surface θ(ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(ω). It is shown that the infinitesimal rigid displacements of the same surface θ(ω) span the tangent space at the origin to this submanifold.
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On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional elasticity
Comptes Rendus Mathématique, 2003Co-Authors: Philippe G. Ciarlet, Cristinel MardareAbstract:Let Ω be an open Connected Subset of R^3 and let Θ be an immersion from Ω into R^3. It is established that the set formed by all rigid displacements of the open set Θ(Ω) is a submanifold of dimension 6 and of class C^∞ of the space H^1(Ω). It is also shown that the infinitesimal rigid displacements of the same set Θ(Ω) span the tangent space at the origin to this submanifold.
Benedikt Steinar Magnand - One of the best experts on this subject based on the ideXlab platform.
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Disc formulas for the weighted Siciak-Zahariuta extremal function
2007Co-Authors: Benedikt Steinar MagnandAbstract:We prove a disc formula for the weighted Siciak-Zahariuta extremal func- tion VX,q for an upper semicontinuous function q on an open Connected Subset X in C n . This function is also known as the weighted Green function with logarithmic pole at infinity and weighted global extremal function. Introduction. If X is a Subset of C n and q : X → R = (−∞, ∞) is a function, then the weighted Siciak-Zahariuta extremal function VX,q with respect to q is defined as
Yannick Privat - One of the best experts on this subject based on the ideXlab platform.
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Actuator design for parabolic distributed parameter systems with the moment method
SIAM Journal on Control and Optimization, 2017Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open Connected Subset Ω of IR n. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in Ω, over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L 2-energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional parabolic operators, but we also provide several examples in higher dimensions. We consider two types of controllers: either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results.
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Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Journal of the European Mathematical Society, 2016Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the wave and Schrödinger equations on a bounded open Connected Subset $\Omega$ 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 $\omega$ of $\Omega$ during a time interval $[0, T]$ with $T>0$. It is well known that, if the pair $(\omega,T)$ satisfies the Geometric Control Condition ($\omega$ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in $\omega \times (0, T)$. We address the problem of the optimal location of the observation Subset $\omega$ among all possible Subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the Subset $\omega$ has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions on $\Omega$, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain $\Omega$). We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures. Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
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Complexity and regularity of maximal energy domains for the wave equation with fixed initial data
Discrete and Continuous Dynamical Systems - Series A, 2015Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:We consider the homogeneous wave equation on a bounded open Connected Subset $\Omega$ of $\R^n$. Some initial data being specified, we consider the problem of determining a measurable Subset $\omega$ of $\Omega$ maximizing the $L^2$-norm of the restriction of the corresponding solution to $\omega$ over a time interval $[0,T]$, over all possible Subsets of $\Omega$ having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of Connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of Connected components.
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Optimal design of sensors for a damped wave equation
2015Co-Authors: Yannick Privat, Emmanuel TrélatAbstract:In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open Connected Subset Ω of IR n. Sensors are modeled by subdomains of Ω of a given measure L|Ω|, with 0 < L < 1. We prove that, if L is close enough to 1, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
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Optimal shape and location of sensors for parabolic equations with random initial data
Archive for Rational Mechanics and Analysis, 2015Co-Authors: Yannick Privat, Emmanuel Trélat, Enrique ZuazuaAbstract:In this article, we consider parabolic equations on a bounded open Connected Subset $\Omega$ of $\R^n$. We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable Subsets of $\Omega$ of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of Connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in [56] for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations.
Ragnar Sigurdsson - One of the best experts on this subject based on the ideXlab platform.
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disc formulas for the weighted siciak zahariuta extremal function
arXiv: Complex Variables, 2006Co-Authors: Benedikt Steinar Magnusson, Ragnar SigurdssonAbstract:We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function $q$ on an open Connected Subset $X$ in $\C^n$. This function is also known as the weighted Green function with logaritmic pole at infinity and weighted global extremal function.