We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with p-growth for p converging to the space dimension n. We prove that for p close to the critical exponent n there are three regimes, two with a non-trivial size of the perforations (exponential and mixed polynomial-exponential) and one where the Γ-limit is alw...

We study Γ-convergence for a sequence of parabolic function-als, F ε (u) = T 0 Ω f (x ε , t, ∇u)dxdt as ε → 0, where the integrand f is nonconvex, and periodic on the first variable. We obtain the representation formula of the Γ-limit. Our results in this paper support a conclusion which relates Γ-convergence of parabolic functionals to the associated gradient flows and confirms one of De Giorg...

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ...

Piecewise constant Mumford-Shah segmentation [17] has been rediscovered by Chan and Vese [6] in the context of region based active contours. The work of Chan and Vese demonstrated many practical applications thanks to their clever numerical implementation using the level-set technology of Osher and Sethian [18]. The current work proposes a Γ -convergence formulation to the piecewise constant Mu...

We study the thin-shell limit of the nonlinear elasticity model for vesicles introduced in part I. We consider vesicles of width 2ε ↓ 0 with elastic energy of order ε 3. In this regime, we show that the limit model is a bending theory for generalized hypersurfaces — namely, co-dimension 1 oriented varifolds without boundary. Up to a positive factor, the limit functional is the Willmore energy. ...

The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50’s, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of Γ-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model ...

In this paper, we establish the convergence of the Ohta–Kawasaki equation to motion by nonlocal Mullins–Sekerka law on any smooth domain in space dimensions N ≤ 3. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are (i) well-preparedness of the initial data and (ii) smoothness of the limiting interface. Ou...

Let I be an open bounded interval of R and W a non-negative continuous function vanishing only at α, β ∈ R. We investigate the asymptotic behaviour in terms of Γ-convergence of the following functional Gε(u) := ε p−2 ∫∫ I×I ∣∣∣∣u(x)− u(y) x− y ∣∣∣∣pdxdy + 1ε ∫ I W (u) dx (p > 2), as ε→ 0. Mathematics Subject Classification (2000). Primary 82B26, 49J45 ; Secondary 49Q20.

We investigate the possibility of describing the “limit problem” of a sequence of optimal control problems (P)(bn), each of which is characterized by the presence of a time dependent vector valued coefficient bn = (bn1 , . . . , bnM ). The notion of “limit problem” is intended in the sense of Γ-convergence, which, roughly speaking, prescribes the convergence of both the minimizers and the infim...