In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are ε-periodic functions modulated by a macroscopic variable, where ε is a small parameter. The mean free path of the particles is also of order ε. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point x 0 where its Hessian matrix is positive definite. This assumption yields a concentration phenomenon around x 0, as ε goes to 0, at a new scale of the order of $\sqrt{\varepsilon}$ which is superimposed with the usual ε oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt{\varepsilon}$ , i.e., of the form $\exp \left (- \frac {1} {2 \varepsilon} M (x-x_0)\cdot (x-x_0) \right )$ , where M is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.


We study a two-dimensional model for micromagnetics, which consists in an energy functional over S 2-valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit. These lower bounds are proved to be optimal and are achieved by one-dimensional profiles, corresponding to Néel walls, if the jump is small enough (less than π/2 in angle), and by two-dimensional profiles, corresponding to cross-tie walls, if the jump is bigger. Thus, it provides an example of a vector-valued phase-transition type problem with an explicit non-one-dimensional energy-minimizing transition layer. We also establish other lower bounds and compactness properties on different quantities which provide a good notion of convergence and cost of vortices.

We introduce an intrinsic notion of perimeter for subsets ofa general Minkowski space (i.e. a finite dimensional Banach space in which thenorm is not required to be even).We prove that this notion of perimeter is equivalent tothe usual definition of surface energy for crystals andwe study the regularity properties of the minimizers and the quasi-minimizers of perimeter.In the two-dimensional case we obtain optimal regularity results:apart from a singular set (which is ${\mathcal H}^1$ -negligible and is empty whenthe unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locallyparameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs.

In this paper we consider dispersive electromagnetic systems in dielectric materials in the presence of acoustic wavefronts. A theory for existence, uniqueness, and continuous dependence on data is presented for a general class of systems which include acoustic pressure-dependent Debye polarization models for dielectric materials.

Let L be a parabolic second order differential operator on the domain $ \bar{\Pi}=\left[ 0,T\right] \times {\mathbb R}.$ Given a function $\hat{u}: {\mathbb R\rightarrow R}$ and $\hat{x}>0$ such that the support of û is contained in $(-\infty ,-\hat{x}]$, we let $\hat{y}:\bar{\Pi}\rightarrow {\mathbb R}$ be the solution to the equation: \[ L\hat{y}=0,\text{\quad }\hat{y}|_{\{0\}\times {\mathbb R}}=\hat{u} . \] Given positive bounds $0<x_{0}<x_{1},$ we seek a function u with support in $\left[ x_{0},x_{1}\right] $ such that the corresponding solution y satisfies: \[ y(t,0)=\hat{y}(t,0)\quad \quad \forall t\in \left[ 0,T\right] . \] We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that $\hat{y}|_{[0,T]\times \{0\}}$ can be C0-approximated, but an exact solution does not exist in general. This result solves the problem of almost replicating a barrier option in the generalised Black–Scholes framework with a combination of European options, as stated by Carr et al. in [6].


The boundary control problem for the dynamical Lame system(isotropic elasticity model) is considered. The continuity ofthe “input → state" map in L 2-norms is established. A structure of thereachable sets for arbitrary T>0 is studied.In general case, only the first component $u(\cdot ,T)$ of thecomplete state $\{ u(\cdot ,T),u_t(\cdot ,T)\}$ may be controlled, an approximate controllability occurring inthe subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundarydata continuation. If T 0 exceeds the time neededfor shear waves to fill the entire domain, then the responseoperator (“input → output" map) $R^{2T_0}$ uniquely determinesRT for any T>0. A procedure recovering R ∞via $R^{2T_0}$ is also described.

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games.We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games.Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.

In this paper we present some applications of the J.-L. Lions' optimal control theory to real life problems in engineering and environmental sciences. More precisely, we deal with the following three problems: sterilization of canned foods, optimal management of waste-water treatment plants and noise control

We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of $\mathbb{R}^N$ , N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context) $E_\varepsilon(u_\varepsilon)\le M_0|\log\varepsilon|$ , where M 0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equation.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class ofnon coercive nonlinear problems whose prototype is
 $$\left\{- \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f\hbox{in}\quad\Omega, u=0\hbox{on} \quad \partial\Omega,\right.$$ 
where Ω is a bounded open subset of ${\mathbb{R}}^N$ , N > 2, 2-1/N < p < N , a belongs to L ∞(Ω), $a(x)\ge\alpha_0>0$ ,f is a function inL 1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space $\xR^n$ ($n\ge2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\,{\rm d}x=c(t)\delta_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

A multiphase generalization of the Monge–Kantorovich optimaltransportation problem is addressed. Existence of optimal solutions is established.The optimality equations are related to classical Electrodynamics.

The goal of this paper is to derive some error estimates for thenumerical discretization of some optimal control problems governedby semilinear elliptic equations with bound constraints on thecontrol and a finitely number of equality and inequality stateconstraints. We prove some error estimates for the optimalcontrols in the L ∞ norm and we also obtain error estimatesfor the Lagrange multipliers associated to the state constraintsas well as for the optimal states and optimal adjoint states.

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.

The purpose of this paper is to show that the method of controlledLagrangians and its Hamiltonian counterpart (based on the notionof passivity) are equivalent under rather general hypotheses. Westudy the particular case of simple mechanical control systems(where the underlying Lagrangian is kinetic minus potentialenergy) subject to controls and external forces in some detail.The equivalence makes use of almost Poisson structures (Poissonbrackets that may fail to satisfy the Jacobi identity) on theHamiltonian side, which is the Hamiltonian counterpart of a classof gyroscopic forces on the Lagrangian side.

Output least squares stability for the diffusion coefficient in an elliptic equation in dimensiontwo is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability.The analysis shows the influence of the flow direction on the parameter to be estimated.A scale analysis for multi-scale resolution of the unknown parameter is provided.

In this paper, we investigate the problem of fast rotatingfluids between two infinite plates with Dirichlet boundary conditions and“turbulentviscosity” for general L 2 initial data. We use dispersive effect toprove strongconvergence to the solution of the bimensionnal Navier-Stokes equationsmodified bythe Ekman pumping term.

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image of a torus. Moreover the global attractor attracts all the solutions of the NS system with exponential rate, that is, the attractor is exponential. We also consider the 2D Navier–Stokes system with rapidly oscillating external force g(x,t,t/ε), which has the average as ε → 0+. We assume that the function g(x,t,z) has a bounded primitive with respect to z and the averaged NS system has a small Grashof number that provides a simple structure of the averaged global attractor. Then we prove that the distance from the global attractor of the original NS system to the attractor of the averaged NS system is less than a small power of ε.

This article is divided into two chapters. Theclassical problem of homogenization of elliptic operators withperiodically oscillating coefficients is revisited in thefirst chapter. Following a Fourier approach, we discuss someof the basic issues of the subject: main convergence theorem,Bloch approximation, estimates on second order derivatives,correctors for the medium, and so on. The second chapter isdevoted to the discussion of some non-classical behaviour ofvibration problems of periodic structures.