Gianni Dal Maso

A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results

Abstract We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G. A. Francfort and J.-J. Marigo, and based on Griffiths theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time-discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.

An existence result for a class of shape optimization problems

Given a bounded open subset Ω of Rn, we prove the existence of a minimum point for a functional F defined on the family A(Ω) of all “quasiopen” subsets of Ω, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(Ω) with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue (or with minimal capacity) with respect to a given elliptic operator.

Ennio De Giorgi

Ennio De Giorgi was born in Lecce on 8 February 1928. His father, Nicola, taught literature at the high school in Lecce, and was an expert in Arabic, history and geography; his mother, Stefania Scopinich, came from a family of seafarers from Losinj in Croatia. His father died prematurely in 1930, but his mother, to whom Ennio was especially close, lived until 1988.

Wiener's criterion and Γ-convergence

Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form −Δu + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ℝn. The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the “energy” of local solutions are obtained.

Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions

We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.

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.

On the relaxation in BV (Ω ; Rm) of quasi-convex integrals

Abstract Given a quasi-convex function f with linear growth, we find the integral representation in BV(Ω; R m ) of the functional F arising from the relaxation of F(u) = ∝ Ω f(▽u) dx, u ϵ C 1 (Ω; R m ) , in the L loc 1 (Ω; R m ) topology.

NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in to the unique solution of the problem When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions of the equations where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

Composite Media and Homogenization Theory

We give an integral representation result for functionals defined on Sobolev spaces; more precisely, for a functional F , we find necessary and sufficient conditions that imply the integral representation formula F (u,B) = ∫

Definition and existence of renormalized solutions of elliptic equations with general measure data

Abstract We introduce a new definition of solution for the nonlinear monotone elliptic problem-div(a(a;, ∇u)) = μ in Ω u = 0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω. We prove the existence of such a solution, a stability result, and partial uniqueness results.