$\Gamma$-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 ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by$${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$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 $W$.

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 ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by $${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$ 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.

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...

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term,...

Soit $\mathbf{A}$ un ensemble quelconque d’opérateurs différentiels en deux variables à coefficients complexes constants. Soit $C_0$ l’espace des fonctions continues complexes tendant vers zéro à l’infini dans le plan euclidien. Soit $C_0\left(\mathbf{A}\right)$ l’espace $\{f:f\in C_0,A_f{C}_0$, tout $a\in \mathbf{A}\}$. Classifier ces espaces équivaut à trouver des conditions nécessaires et suffisantes sur des opérateurs différentiels $P_1,...,P_n$ pour que $\parallel P_1\phi {\parallel }_{\infty }\le K(\parallel P_2\phi {\parallel }_{\infty }+\cdots +\parallel P_n\phi {\parallel }_{\infty })$. Il paraît que ce problème général est bien difficile. Nous présentons ici la solution complète dans le cas spécial des $C_0\left(\mathbf{A}\right)$ stables...

If $P$ is a polynomial in ${\mathbf{R}}^n$ such that $1/P$ integrable, then the inverse Fourier transform of $1/P$ is a fundamental solution $E_P$ to the differential operator $P\left(D\right)$. The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$. For $n=1$ it is shown that $E_P$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$. The singularities of this extension are studied.

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized...

Integral representation of relaxed energies and of Γ-limits of functionals $$(u,v)\mapsto {\int }_{\Omega }f(x,u\left(x\right),v\left(x\right))dx$$ 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.

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover,...