Homogenization of networks in domains with oscillating boundaries

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 $\Gamma$-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, respectivel

Discrete double-porosity models for spin systems

We consider spin systems between a finite number N of “species” or “phases” partitioning a cubic lattice Zd . We suppose that interactions between points of the same phase are coercive while those between points of different phases (or possibly between points of an additional “weak phase”) are of lower order. Following a discrete-to-continuum approach, we characterize the limit as a continuum energy defined on N-tuples of sets (corresponding to the N strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part that describes the combined effect of lowerorder terms, weak interactions between phases, and possible oscillations in the weak phase

Discrete double-porosity models for spin systems

We consider spin systems between a finite number N of “species” or “phases” partitioning a cubic lattice Zd . We suppose that interactions between points of the same phase are coercive while those between points of different phases (or possibly between points of an additional “weak phase”) are of lower order. Following a discrete-to-continuum approach, we characterize the limit as a continuum energy defined on N-tuples of sets (corresponding to the N strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part that describes the combined effect of lowerorder terms, weak interactions between phases, and possible oscillations in the weak phase

Homogenization of discrete high-contrast energies

Abstract. This paper focuses on deriving double-porosity models from simple high-contrast atomistic interactions. Using the variational approach and Γ-convergence techniques we derive the effective double-porosity type problem and prove the convergence. We also consider the dynamical case and study the asymptotic behavior of solutions for the gradient flow of the corresponding discrete functionals

Spectral gaps for the linear water-wave problem in a channel with thin structures

Peer reviewe

Another brick in the wall

We study the homogenization of a linearly elastic energy defined on a periodic collection of disconnected sets with a unilateral condition on the contact region between two such sets, with the model of a brick wall in mind. Using the language of Gamma-convergence we show that the limit homogenized behavior of such an energy can be described on the space of functions with bounded deformation using the masonry-type functionals studied by Anzellotti, Giaquinta and Giusti. In this case, the limit energy density is given by the homogenization formula related to the brick-wall type energy