Anneliese Defranceschi

G-convergence of monotone operators

Abstract A general notion of G-convergence for sequences of maximal monotone operators of the form is introduced in terms of the asymptotic behavior, as h → + ∞, of the solutions u h to the equations and of their momenta a h ( x , D u h ). The main results of the paper are the local character of the G-convergence and the G-compactness of some classes of nonlinear monotone operators.

Limits of nonlinear Dirichlet problems in varying domains

We study the general form of the limit, in the sense of Γ-convergence, of a sequence of nonlinear variational problems in varying domains with Dirichlet boundary conditions. The asymptotic problem is characterized in terms of the limit of suitable nonlinear capacities associated to the domains.

Homogenization of almost periodic monotone operators

Abstract We determine some sufficient conditions for the G-convergence of sequences of quasi-linear monotone operators, together with an asymptotic formula for the G-limit. We then prove a homogenization theorem for quasiperiodic monotone operators and, eventually, extend this result to general almost periodic monotone operators using an approximation result and a closure lemma.

Limits of minimum problems with convex

This work deals with the asymptotic behaviour of a sequence of minimum problems on a Sobolev space of vector valued functions subject to constraints of obstacle type. We consider sequences of the formMin{ n W ( x , Du(z)) dr : u e E H1> O,(Ω Rm), u(x) e Kn ( x ) for p.e. x ´ A}, (∗) where Ω is an open subset of Rn, m 2 1, W is quadratic in the second variable and non-negative, A is an open subset of R, and Kh (h ´ N) is a closed and convex valued multifunction from Ω to Rm.The well-known relaxation phenomenon of the scalar case still takes place for (∗); this is obtained by proving a compactness result for a general class of constraint functionals. Applications are given to Dirichlet problems in perforated domains for the usual energy functional of linearized elasticity.

Convergence of unilateral problems for monotone operators

We prove some new results on the convergence of variational inequalities for monotone operators, when both the operator and the obstacle are perturbed.

Variational evolution of one-dimensional Lennard-Jones systems

We analyze Lennard-Jones systems from the standpoint of variational principles beyond the static framework. In a one-dimensional setting such systems have already been shown to be equivalent to energies of Fracture Mechanics. Here we show that this equivalence can also be given in dynamical terms using the notion of minimizing movements.

A characterization ofC 1-convex sets in Sobolev spaces

In this paper we prove that, ifK is a closed subset ofW01,p (Ω,Rm) with 1<p<+∞ andm≥1, thenK is stable under convex combinations withC1 coefficients if and only if there exists a closed and convex valued multifunction from Ω toRm such thatThe casem=1 has already been studied by using truncation arguments which rely on the order structure ofR (see [2]). In the casem>1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed.Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type.

Homogenization of quasi-linear equations with natural growth terms

AbstractIn this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form

Homogenization of multiple integrals

- div\left( {a\left( {\tfrac{x}{\varepsilon },Du_\varepsilon } \right)} \right) + \gamma \left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon = H\left( {\tfrac{x}{\varepsilon },u_\varepsilon ,Du_\varepsilon } \right) + h(x)