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Marcello Ponsiglione - One of the best experts on this subject based on the ideXlab platform.
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A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Alessandro Giacomini, Marcello PonsiglioneAbstract:We prove a stability result for a large class of unilateral Minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form where S ( u ) is the jump set of u and is a sequence of rectifiable sets with We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for ( N −1)-rectifiable sets called σ -convergence, which is useful in the study of the stability of unilateral Minimality properties.
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A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Alessandro Giacomini, Marcello PonsiglioneAbstract:We prove a stability result for a large class of unilateral Minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form Open image in new window where S(u) is the jump set of u and Open image in new window is a sequence of rectifiable sets with Open image in new window We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for (N−1)-rectifiable sets called σ-convergence, which is useful in the study of the stability of unilateral Minimality properties.
Alessandro Giacomini - One of the best experts on this subject based on the ideXlab platform.
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A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Alessandro Giacomini, Marcello PonsiglioneAbstract:We prove a stability result for a large class of unilateral Minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form where S ( u ) is the jump set of u and is a sequence of rectifiable sets with We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for ( N −1)-rectifiable sets called σ -convergence, which is useful in the study of the stability of unilateral Minimality properties.
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A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Alessandro Giacomini, Marcello PonsiglioneAbstract:We prove a stability result for a large class of unilateral Minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form Open image in new window where S(u) is the jump set of u and Open image in new window is a sequence of rectifiable sets with Open image in new window We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for (N−1)-rectifiable sets called σ-convergence, which is useful in the study of the stability of unilateral Minimality properties.
Massimiliano Morini - One of the best experts on this subject based on the ideXlab platform.
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Isoperimetry and stability properties of balls with respect to nonlocal energies
Communications in Mathematical Physics, 2014Co-Authors: Alessio Figalli, Nicola Fusco, Francesco Maggi, Vincent Millot, Massimiliano MoriniAbstract:We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global Minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal s-perimeter plus a nonlocal repulsive interaction term. In the particular case s = 1 the s-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov [25, 26] concerning Minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.
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Minimality via second variation for a nonlocal isoperimetric problem
Communications in Mathematical Physics, 2013Co-Authors: Emilio Acerbi, Nicola Fusco, Massimiliano MoriniAbstract:We discuss the local Minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L 1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local Minimality of certain lamellar configurations.
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equilibrium configurations of epitaxially strained elastic films second order Minimality conditions and qualitative properties of solutions
Archive for Rational Mechanics and Analysis, 2012Co-Authors: Nicola Fusco, Massimiliano MoriniAbstract:We consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present. We study quantitative and qualitative properties of equilibrium configurations, that is, of local and global minimizers of the free-energy functional. More precisely, we determine analytically the critical threshold for the local Minimality of the flat configuration and we also prove several results concerning its global Minimality. The non-occurrence of singularities in non-flat global minimizers is also addressed. One of the main results of the paper is a new sufficient condition for local Minimality, which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies.
W.s. Gray - One of the best experts on this subject based on the ideXlab platform.
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Minimality and local state decompositions of a nonlinear state space realization using energy functions
IEEE Transactions on Automatic Control, 2000Co-Authors: J.m.a. Scherpen, W.s. GrayAbstract:In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state-space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies Minimality. Using the nonlinear analogue of the Kalman decomposition, connections are then established between Minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.
Carolyn L. Beck - One of the best experts on this subject based on the ideXlab platform.
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On formal power series representations for uncertain systems
IEEE Transactions on Automatic Control, 2001Co-Authors: Carolyn L. BeckAbstract:The concept of Minimality as developed for uncertain and multidimensional systems represented by linear fractional transformations (LFTs) is related to realization theory results for formal power series. We discuss the relationship between the notions of Minimality for LFT and series realizations, and present a method for obtaining one type of minimal realization from the opposing type. An extension of an existing Minimality result for formal power series to the multi-input-multi-output (MIMO) case is also presented.