G. Dal Maso

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and Γ-convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (Ph) of optimal control problems and a control problem (P∞), we determine some general conditions, involvingG-convergence and Γ-convergence, under which the sequence of the optimal pairs of the problems (Ph) converges to the optimal pair of problem (P∞).

New lower semicontinuity results for polyconvex integrals

SummaryWe study integral functionals of the formF(u, Ω)=∫Ωf(▽u)dx, defined foru ∈ C1(Ω;Rk), Ω⊑Rn. The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) ≥c0¦ℳ(A)¦ for a suitable constant c0 > 0, where ℳ(A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC1(Ω;Rk) with respect to the strong topology ofL1(Ω;Rk). Then we consider the relaxed functional ℱ, defined as the greatest lower semicontinuous functional onL1(Ω;Rk) which is less than or equal toF on C1(Ω;Rk). For everyu ∈ BV(Ω;Rk) we prove that ℱ (u,Ω) ≥ ∫Ωf(▽u)dx+c0¦Dsu¦(Ω), whereDu=▽u dx+Dsu is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that ℱ (u,Ω)=∫Ωf(▽u)dx for everyu ∈ W1,p(Ω;Rk), withp ≥ min{n,k}. We prove also that the functional ℱ (u, Ω) can not be represented by an inte- gral for an arbitrary functionu ∈ BVloc(Rn;Rk). In fact, two examples show that, in general, the set functionΩ → ℱ (u, Ω) is not subadditive whenu ∈ BVloc(Rn;Rk), even ifu ∈ Wloc1,p(Rn;Rk) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu ∈ BV(Ω;Rk) such that ℱ (u, Ω)=∫Ωf(▽u)dx, particularly in the model casef(A)=¦ℳ(A)¦.

Linearized Elasticity as Γ-Limit of Finite Elasticity

Linearized elastic energies are derived from rescaled nonlinear energies by means of Γ-convergence. For Dirichlet and mixed boundary value problems in a Lipschitz domain Ω, the convergence of minimizers takes place in the weak topology of H1(Ω,Rn) and in the strong topology of W1,q(Ω,Rn) for 1≤q<2.

On a class of optimum problems in structural design

The paper deals with the existence of solutions for a class of optimal design problems. The notion of relaxation of an integral functional with respect toG-convergence is introduced, and a general integral representation theorem is obtained for the relaxed functional. For a particular class of functionals, this integral representation is computed explicitly.

Dirichlet problem for demi-coercive functionals

On considere des fonctionnelles integrales du type F(u)=∫ Ω f(Du), ou f:R N →[0,+∞] est une fonction convexe ayant la propriete suivante: il existe a>0, b≥0, γ∈R N tel que f(z)≥a|z|− −b ∀z∈R N

Variational inequalities for the biharmonic operator with variable obstacles.

SummaryThe asymptotic behaviour of the solutions of a sequence of variational inequalities for the biharmonic operator with variable two-sided obstacles is investigated by describing the form of the limit problem, which is computed explicitly in two meaningful examples.