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3D-2D asymptotic analysis for micromagnetic thin films

Roberto Alicandro, Chiara Leone (2001)

ESAIM: Control, Optimisation and Calculus of Variations

Γ -convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure Ω ε = ω × ( - ε , ε ) , ω 2 , whose energy is given by ε ( m ¯ ) = 1 ε Ω ε W ( m ¯ , m ¯ ) + 1 2 u ¯ · m ¯ d x subject to div ( - u ¯ + m ¯ χ Ω ε ) = 0 on 3 , and to the constraint | m ¯ | = 1 on Ω ε , where W is any continuous function satisfying p -growth assumptions with p > 1 . Partial results are also obtained in the case p = 1 , under an additional assumption on W .

3D-2D Asymptotic Analysis for Micromagnetic Thin Films

Roberto Alicandro, Chiara Leone (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure Ω ε = ω × ( - ε , ε ) , ω 2 , whose energy is given by ε ( m ¯ ) = 1 ε Ω ε W ( m ¯ , m ¯ ) + 1 2 u ¯ · m ¯ d x subject to div ( - u ¯ + m ¯ χ Ω ε ) = 0 on 3 , and to the constraint | m ¯ | = 1 on Ω ε , where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

517.98

A.M. Вершик (1984)

Zapiski naucnych seminarov Leningradskogo

Γ -convergence and absolute minimizers for supremal functionals

Thierry Champion, Luigi De Pascale, Francesca Prinari (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the L p approximants naturally associated to a supremal functional Γ -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local...

Γ -convergence of concentration problems

Micol Amar, Adriana Garroni (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we use Γ -convergence techniques to study the following variational problem S ε F ( Ω ) : = sup ε - 2 * Ω F ( u ) d x : Ω | u | 2 d x ε 2 , u = 0 on Ω , where 0 F ( t ) | t | 2 * , with 2 * = 2 n n - 2 , and Ω is a bounded domain of n , n 3 . We obtain a Γ -convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem S ε F ( Ω ) . Finally, a second order expansion in Γ -convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

Γ -convergence of constrained Dirichlet functionals

Gian Paolo Leonardi (2003)

Bollettino dell'Unione Matematica Italiana

Given an open, bounded and connected set Ω R n with Lipschitz boundary and volume Ω , we prove that the sequence F k of Dirichlet functionals defined on H 1 Ω ; R d , with volume constraints v k on m 2 fixed level-sets, and such that i = 1 m v i k < Ω for all k , Γ -converges, as v k v with i = 1 m v i k = Ω , to the squared total variation on B V V ; R d , with v as volume constraint on the same level-sets.

Γ -convergence of discrete approximations to interfaces with prescribed mean curvature

Giovanni Bellettini, Maurizio Paolini, Claudio Verdi (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The numerical approximation of the minimum problem: min A Ω F ~ A , is considered, where F ~ A = P Ω A + cos θ H n - 1 A Ω - A κ . The solution to this problem is a set A Ω R n with prescribed mean curvature κ and contact angle θ at the intersection of A with Ω . The functional F ~ is first relaxed with a sequence of nonconvex functionals defined in H 1 Ω which, in turn, are discretized by finite elements. The Γ -convergence of the discrete functionals to F ~ as well as the compactness of any sequence of discrete absolute minimizers are proven.

Γ Limiti e analisi non standard

Vincenzo M. Tortorelli (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this note we give a nonstandard characterization of multiple topological Γ operators as sup-min of standard part map.

Γ -limiti e minimi di Pareto

Roberto Peirone (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The notion of Γ -limit is extended from the case of functions with values in 𝐑 ¯ to the case of those with values in an arbitrary complete lattice and the problem of convergence of Pareto minima related to a convex cone is considered.

Γ-convergence and absolute minimizers for supremal functionals

Thierry Champion, Luigi De Pascale, Francesca Prinari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove that the Lp approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution)...

Γ-convergence of functionals on divergence-free fields

Nadia Ansini, Adriana Garroni (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions.

Γ-limits of convolution functionals

Luca Lussardi, Annibale Magni (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We compute the Γ-limit of a sequence of non-local integral functionals depending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density.

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