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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

A compactness result for a second-order variational discrete model

Andrea Braides, Anneliese Defranceschi, Enrico Vitali (2012)

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

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower...

A compactness result for a second-order variational discrete model

Andrea Braides, Anneliese Defranceschi, Enrico Vitali (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions...

A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds

Annibale Magni (2015)

Geometric Flows

We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...

A general Hamilton-Jacobi framework for non-linear state-constrained control problems

Albert Altarovici, Olivier Bokanowski, Hasnaa Zidani (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...

A generalized dual maximizer for the Monge–Kantorovich transport problem

Mathias Beiglböck, Christian Léonard, Walter Schachermayer (2012)

ESAIM: Probability and Statistics

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y →  [0,∞]  is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel’s perturbation technique.

A generalized dual maximizer for the Monge–Kantorovich transport problem∗

Mathias Beiglböck, Christian Léonard, Walter Schachermayer (2012)

ESAIM: Probability and Statistics

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y →  [0,∞]  is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic...

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

Deville, Robert (1995)

Serdica Mathematical Journal

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition....

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