A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. Under convexity and p  -growth conditions (p>1$p>1$), it is proved that all such possible effective energy densities obtained by a Γ-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.     

Suboptimal boundary controls for elliptic equation in critically perforated domain

In this paper we study the asymptotic behaviour, as ε   tends to zero, of a class of boundary optimal control problems Pε${\mathbb{P}}_{\epsilon }$, set in ε-periodically perforated domain. The holes have a critical size with respect to ε  -sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type; on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problems Pε${\mathbb{P}}_{\epsilon }$.     

Quasistatic crack growth in finite elasticity with non-interpenetration

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.     

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.      

Homogenization of a parabolic model of ferromagnetism

This work deals with the homogenization of hysteresis-free processes in ferromagnetic composites. A degenerate, quasilinear, parabolic equation is derived by coupling the Maxwell-Ohm system without displacement current with a nonlinear constitutive law:

     

Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.     

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p$p$-Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ$\lambda$ is greater than the second eigenvalue _λ2${\lambda }_2$ of the negative p$p$-Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p$p$-Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.     

Reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media

We present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients, which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures. Moreover, we study the Γ-convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence-free fields applying the properties of multiscale Young measures associated with sequences of divergence-free fields.     

Limiting properties of the second order Ginzburg–Landau minimizers

The author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0. And this paper is also concerned with the location of the zeros and the blow-up points of the gradient of the minimizers of this functional. Finally, the strong convergence of the gradient of the radial minimizers is obtained.     

Some homogenization results for non-coercive Hamilton–Jacobi equations

Recently, C. Imbert and R. Monneau study the homogenization of coercive Hamilton–Jacobi Equations with a u/ε-dependence: this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for “standard” Hamilton–Jacobi Equations (i.e. without a u/ε-dependence) but in the case of non-coercive Hamiltonians. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert and R. Monneau, but under slightly more restrictive assumptions on the Hamiltonians.     

Young measures, weak and strong convergence and the Visintin-Balder Theorem

This paper is concerned with sequences inL ¹ which converge weakly. Young's measures theory permits us to give sufficient conditions insuring the strong convergence and to understand the behaviour of the sequences which do not converge strongly.     

On the existence of some new positive interior spike solutions to a semilinear Neumann problem

In this paper we are concerned with the following Neumann problem

     

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.     

A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy

We study a quasistatic evolution problem for a nonconvex elastic energy functional. Due to lack of convexity, the natural energetic formulation can be obtained only in the framework of Young measures. Since the energy functional may present multiple wells, an evolution driven by global minimizers may exhibit unnatural jumps from one well to another one, which overcome large potential barriers. To avoid this phenomenon, we study a notion of solution based on a viscous regularization. Finally we compare this solution with the one obtained with global minimization.     

Asymmetric potentials and motor effect: a homogenization approach

We provide a mathematical analysis for the appearance of motor effects, i.e., the concentration (as Dirac masses) at one side of the domain, for the solution of a Fokker–Planck system with two components, one with an asymmetric potential and diffusion and one with pure diffusion. The system has been proposed as a model for motor proteins moving along molecular filaments. Its components describe the densities of different conformations of proteins.     

Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary

We study the convergence of the solutions of the Dirichlet problem associated to a degenerate nonlinear higher-order elliptic equations in divergence form in variables domains, to a limit solution of the same type problem in a fixed domain, following the methods of the asymptotic expansion developed by Skrypnik [Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, AMS, Providence, RI, 1994] modified to weighted higher-order case.     

Second-order energies on thin structures: variational theory and non-local effects

For a given positive measure μ on , we consider integral functionals of the kind

     

A comparison result for elliptic equations in the infinite dimensional Gauss space

We obtain a comparison result for a class of Dirichlet problems for the operator in an infinite dimensional separable Hilbert space X with the Gauss measure γ and a suitable differentiable structure.