Asymptotic Analysis of Periodically-Perforated Nonlinear Media at the Critical Exponent

Abstract

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with n-growth where n is the space dimension, showing that there exists a critical scale for the perforations such that the Γ-limit is non-trivial. We prove that the limit extra-term is given by a formula of homogenization type, which simplifies in the case of n-homogeneous energy densities.

Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent

“…This can be done explicitly if K is a ball, and gives (6) as a result. Note that in this case the radius of K does not affect the result; we can therefore extend the result to arbitrary K with nonempty interior by comparison with the case of balls containing K or contained in K, respectively, and conclude that the form of the limit is indeed independent of the shape of K. Further technical arguments are needed when f is not positively homogeneous; a detailed proof can be found in [9].…”

“…Our arguments show that the exponential regime derives from the scaling invariance of the problems in (9), which eliminates the pre-factor ε n−p , and from the logarithmic behaviour of minimizers. We have shown that the usual 'capacitary' formula for the limit integrand ϕ in the case p < n is substituted by an interesting 'homogenization' formula.…”

“…In the case p = n the same argument gives a trivial lower bound since the corresponding limit computation of the n-capacity of K (with respect to R n ) gives, inf R n |Dv| n dy: v = 0 on K, 1 − v ∈ W 1,n (R n ) = 0, from which we deduce that the limit of the right-hand side of (9) is 0. We have therefore to depart from the proof in [2] by a more difficult analysis of the behaviour of the energies defined by the minimum problems in (9). This can be done explicitly if K is a ball, and gives (6) as a result.…”

“…The proof of this result relies on adapting the arguments in [2] to the case of varying exponent p, to carefully estimate the minimum problems in (9), and understand their interplay with the (unknown) prefactor ε n−p . A detailed proof will appear in [10].…”

Models of defects in atomistic systems

“…To obtain case (2), we first treat the case of K a single hyperplane and ϕ(x, z) = c 1 + c 2 z 2 . This can be obtained following arguments similar to those by Ansini [4] to approximate the energy density c(u + − u − ) 2 on a surface (Neumann sieve) coupled with the description of the effect of pinning sites at the critical scaling by Sigalotti [26,27]. Note that the computation of the interfacial energy gives the same constant as in the continuous case for n = 2, while it highlights a more complex behavior for n ≥ 3, where a fraction of the total contribution is actually given by the strong springs at the interface, which sums up to the contribution distributed away from the interface and summarized in a capacitary formula.…”