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We prove norm-resolvent and spectral convergence in L² of solutions to the Neumann Poisson problem --Δε= f on a domain Ωε perforated by Dirichlet holes and shrinking to a 1- dimensional interval. The limit u satisfies an equation of the type --u" + μu = f on the interval (0, 1), where μ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant μ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation. [ABSTRACT FROM AUTHOR]

A first order model for the transmission of waves through a sound-hard perforation along an interface is derived. Mathematically, we study the Neumann problem for the Helmholtz equation in a complex geometry, the domain contains a periodic array of inclusions of size ε > 0 along a co-dimension 1 manifold. We derive effective equations that describe the limit as ε → 0. At leading order, the Neumann sieve perforation has no effect. The corrector is given by a Helmholtz equation on the unperturbed domain with jump conditions across the interface. The corrector equations are derived with unfolding methods in L 1 -based spaces. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a -limit with respect to an appropriate convergence, is defined in a 'stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for p-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively. [ABSTRACT FROM AUTHOR]

We consider graphs parameterized on a portion X ⊂ Z d × { 1 , ... , M } k of a cylindrical subset of the lattice Z d × Z k , and perform a discrete-to-continuum dimension-reduction process for energies defined on X of quadratic type. Our only assumptions are that X be connected as a graph and periodic in the first d -directions. We show that, upon scaling of the domain and of the energies by a small parameter ɛ , the scaled energies converge to a d -dimensional limit energy. The main technical points are a dimension-reducing coarse-graining process and a discrete version of the p -connectedness approach by Zhikov. [ABSTRACT FROM AUTHOR]

Abstract: We consider the homogenization of a time-dependent heat transfer problem in a thin domain of thickness e in whose heat capacity and conductivity vary periodically in the cross-section with periodicity cell of size . The structure is highly heterogeneous so that the conductivity is of order 1 in two connected parts of the basic cell, separated by a third one having a much lower conductivity of the scale . We assume that e (resp. ) tends to zero with a rate (resp. ). The heat capacities of the three components are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical values of the problem are and , and we identify the homogenized problem depending on whether and are zero, strictly positive finite or infinite. [Copyright &y& Elsevier]

In this contribution the effective material behaviour of thin structures like membranes and plates consisting of heterogeneities is investigated. The diameter of typical inclusions can be in the order of magnitude of the thickness of the membrane or plate. Thus, the prerequisite for a standard homogenization procedure using representative volume elements is not fulfilled anymore, since the required size of an RVE would be larger than the thickness of the structure itself. Additionally the assumption of uniformity of the boundary conditions on the RVE would be violated especially for plates since in general the predominant deformation of such structures is bending. However, it can be shown that the effective behaviour of such heterogeneous thin structures subjected to finite deformations is still in good agreement to the results obtained for homogenized materials. [ABSTRACT FROM AUTHOR]

For a given amount m of mass, we study the class of materials which can be reached by homogenization distributing the mass m on periodic structures of prescribed dimension k ⩽ n in R n . Both in the scalar case of conductivity and in the vector-valued case of elasticity, we find some bounds for the effective coefficients, depending on the mass m and the dimension parameters k , n . In the scalar case we prove that such bounds are optimal, as they do describe the set of all materials reachable by homogenization of structures of the type under consideration; in the vector-valued case we show that some of our estimates are attained. [ABSTRACT FROM AUTHOR]

To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them with a periodic positive measure μ Rn. We introduce a new notion of two-scale convergence for a sequence of functions vε ∈ Lμεp (Ω; Rd where Ω is an open bounded subset of Rn, and the measures με are the ε-scalings of μ namely, με (B) := εn μ(ε-1 B). Enforcing the concept of tangential calculus with respect to measures and related periodic Sobolev spaces, we prove a structure theorem for all the possible two-scale limits reached by the sequences (uε, ∇ uε when {uε} ⊂ C01 (Ω) satisfy the boundedness condition supε ∫Ω |uε|p + |∇uε|p dμε < +∞ and when the measure μ satisfies suitable connectedness properties. This leads us to deduce the homogenized density of a sequence of energies of the form ∫Ω j(x/ε, ∇u) dμε, where j(y,z) is a convex integrand, periodic in y, and satisfying a p-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limit process. [ABSTRACT FROM AUTHOR]