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We prove that for any connected open set Ω ⊂ R n and for any set of matrices K = {A 1 , A 2 , A 3 } ⊂ M m×n , with m ≥ n and rank(A i − A j) = n for i = j, there is no non-constant solution B ∈ L ∞ (Ω, M m×n), called exact solution, to the problem DivB = 0 in D ′ (Ω, R m) and B(x) ∈ K a.e. in Ω. In contrast, A. Garroni and V. Nesi [10] exhibited an example… (More)

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and De-launay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit… (More)

We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider inital data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two… (More)

We prove that rank-.n 1/ convexity does not imply quasiconvexity with respect to divergence free fields (so-called S-quasiconvexity) in M mn for m > n, by adapting the well-known Šverák's counterexample to the solenoidal setting. On the other hand, we also remark that rank-.n 1/ convexity and S-quasiconvexity turn out to be equivalent in the space of n n… (More)

We fully characterize quasiconvex hulls for three arbitrary solenoidal (divergence free) wells in dimension three. With this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic two-dimensional planes and convex in (up to three) rank-2 directions within the planes. Within the framework of the theory of… (More)

- GRÉGOIRE ALLAIRE, MARIAPIA PALOMBARO, JEFFREY RAUCH, Ennio De Giorgi
- 2008

We study, for times of order 1/ε, solutions of wave equations which are O(ε 2) modulations of an ε periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order ε. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity… (More)

We study a partial differential relation that arises in the context of the Born–Infeld equations (an extension of Maxwell's equations) by using Gromov's method of convex integration in the setting of divergence-free fields.

We give a direct proof that the group velocities of Bloch wave packet solutions of periodic second order wave equations cannot exceed the maximal speed of propagation of the periodic wave equation. Dedicated to our friend and colleague J. P. Dias celebrating years of comradery. 1. Main result This note complements our studies [2], [3] of Bloch wave packets.… (More)

We discuss an atomistic model for heterogeneous nanowires, allowing for dislocations at the interface. We study the limit as the atomic distance converges to zero, considering simultaneously a dimension reduction and the passage from discrete to continuum. Employing the notion of Gamma-convergence, we establish the minimal energies associated to defect-free… (More)

We prove that rank-(n−1) convexity does not imply S-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in M m×n for m > n, by adapting the well-knowň Sverák's counterexample [5] to the solenoidal setting. On the other hand, we also remark that rank-(n − 1) convexity and S-quasiconvexity turn out to be equivalent in the space of n ×… (More)