Luciano Carbone

Homogenization with unbounded constraints on the gradient

where Q is an open set. 9 is a suitable functional space for which the variational problem (1.2) with given f makes sense (this is certainly the case if some additional requirements on q and fare fullfilled). Then, according to a conjecture of Bensoussan, Lions and Papanicolaou [3], the sequence uh is expected to converge (in a suitable topology) to a function u. which is solution of a problem of the same type as (1.2), but with a g independent of x (this is the reason for the name of problem of homogenization). Previous papers by several authors have dealt with bounded constraint q, but the problem of unbounded Q, has been settled only in the unidimensional case. For an account of the bibliography, we refer to [lo]. In this paper, we shall assume: (i) 36 E [0, l/2) such that

Some remarks on a problem of homogenization with fixed traces

The problem of the convergence of the solutions of variational problems with constraints on the- gradient and prescribed traces on δΩ is studied. Expected results are proved under “reasonable” compatibility conditions between the constraints and the traces.

An approach to the homogenization of nonlinear elastomers via the theory of unbounded functionals

Abstract The homogenization process for some energies of integral type arising in the modelling of rubber-like elastomers is carried out. The main feature of the variational problems taken into account is the presence of pointwise oscillating constraints on the gradients of the admissible deformations. The classical homogenization result is established also in this framework, both for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained. An explicit computation for the homogenized integrand relative to energy density in a particular relevant case is derived.

Fin junction of ferroelectric thin films

Abstract In this paper, starting from a non-convex and nonlocal 3D-variational model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two orthogonal joined ferroelectric thin films. We obtain three different 2D-variational models for joined thin films, depending on how the reduction happens. Indeed, a memory effect of the reduction process appears, and it depends on the competition of the relative thickness of the two films. The guide parameter is the limit of the ratio between these two small thickness.

Elements of Convex Analysis

Proposition 2.0 (i) Any set K ⊂ B is closed and convex iff IK is closed and convex. (ii) Any function f : B →]−∞,+∞] is lower semicontinuous and convex iff epi(f) is closed and convex. (iii) If {Ki}i∈I is a family of closed convex subsets of B, then ⋂ iKi is closed and convex. (iv) If {fi}i∈I is a family of lower semicontinuous convex functions B →] −∞,+∞], then their upper hull f(·) := supi fi(·) is lower semicontinuous and convex.

Γ-Convergence of Integral Functionals Defined on Vector-Valued Functions

Let (U, τ) be a topological space satisfying the first countability axiom and let F h , h = 1,2,…, be real functionals defined on U.