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3D-2D asymptotic analysis for micromagnetic thin films

ESAIM: Control, Optimisation and Calculus of Variations

$\Gamma$-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon$ approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)$, $\omega \subset {ℝ}^{2}$, whose energy is given by${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$subject to$\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\text{on}{ℝ}^{3},$and to the constraint$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$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

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 ${\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)$, $\omega \subset {ℝ}^{2}$, whose energy is given by ${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ subject to $\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{ℝ}^{3},$ and to the constraint $|\overline{m}|=1\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\Omega }_{\epsilon },$ 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.

A general approximation theorem of Whitney type.

RACSAM

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...

A-Quasiconvexity: Relaxation and Homogenization

ESAIM: Control, Optimisation and Calculus of Variations

Integral representation of relaxed energies and of Γ-limits of functionals $\left(u,v\right)↦{\int }_{\Omega }f\left(x,u\left(x\right),v\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}dx$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p, are recovered.

Bases in solution sheaves of systems of partial differential equations.

Journal für die reine und angewandte Mathematik

Boundary value problems for linear partial differential equation with constant coefficients. Homogeneous equation in the half plane

Časopis pro pěstování matematiky

Caractérisation des problèmes mixtes hyperboliques bien posés

Mémoires de la Société Mathématique de France

Constant Coefficient Differential Operators with Slowly Spreading Solutions.

Mathematische Annalen

Dimension reduction for functionals on solenoidal vector fields

ESAIM: Control, Optimisation and Calculus of Variations

We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 &lt; p &lt; ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint...

Dimension reduction for functionals on solenoidal vector fields

ESAIM: Control, Optimisation and Calculus of Variations

We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint...

Interpolation problems in cones. Nota I

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

In questa nota, si studiano problemi di interpolazione per varietà discrete in spazi di funzioni olomorfe in coni. In particolare si mostra come sia possibile estendere il Principio Fondamentale di Ehrenpreis ad equazioni di convoluzione nella spazio $H_{c}(\Omega)$, introdotto in [4] in connessione con problemi di fisica quantistica.

Interpolation problems in cones. Nota II

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

Si estendono qui i risultati della nota precedente al caso di varietà non discrete. Ciò viene utilizzato per ottenere un teorema di rappresentazione per soluzioni di sistemi di equazioni di convoluzione in spazi di funzioni olomorfe in coni.

On continuation of regular solutions of linear partial differential equations

Banach Center Publications

On the Existence of Real Analytic Solutions of Partial Differential Equations with Constant Coefficients.

Inventiones mathematicae

On variational approach to the Hamilton-Jacobi PDE

Commentationes Mathematicae Universitatis Carolinae

In this paper we construct a minimizing sequence for the problem (1). In particular, we show that for any subsolution of the Hamilton-Jacobi equation $\left(*\right)$ there exists a minimizing sequence weakly convergent to this subsolution. The variational problem (1) arises from the theory of computer vision equations.

Ouverts stablement convexes par rapport à un opérateur différentiel

Annales de l'institut Fourier

On montre l’équivalence entre certaines inégalités “à la Carleman” et certaines propriétés de régularité des solutions à support compact d’équations aux dérivées partielles à coefficients constants : $P\left(D\right)$ étant un opérateur différentiel sur ${\mathbf{R}}^{n}$, on en déduit une caractérisation, en termes d’inégalités ${L}^{2}$, des ouverts $\Omega$ de ${\mathbf{R}}^{n}$ tels que $\Omega ×{\mathbf{R}}^{k}$ soit $P\left(D\right)$-convexe pour tout entier $k$.

Problème de Dirichlet pour des opérateurs hyperboliques de type positif

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Prolongement des solutions analytiques réelles d'équations aux dérivées partielles à coefficients constants

Séminaire Équations aux dérivées partielles (Polytechnique)

Propriétés asymptotiques des solutions d'équations simplement caractéristiques

Séminaire Équations aux dérivées partielles (Polytechnique)

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in ${ℂ}^{N}$

Studia Mathematica

$G\subset {ℂ}^{N}$ is compact and convex it is known for a long time that the nonzero constant coefficients linear partial differential operators (of finite or infinite order) are surjective on the space of all analytic functions on G. We consider the question whether solutions of the inhomogeneous equation can be given in terms of a continuous linear operator. For instance we characterize those sets G for which this is always the case.

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