Calculus of Variations and Geometric Measure Theory

A. Braides - V. Chiadò Piat

Homogenization of networks in domains with oscillating boundaries

created by braidesa on 20 Oct 2017
modified on 21 Jan 2019


Published Paper

Inserted: 20 oct 2017
Last Updated: 21 jan 2019

Journal: Applicable Analysis
Volume: 98
Pages: 45-63
Year: 2019
Doi: 10.1080/00036811.2018.1430782

Special Issue in memory of V.V.Zhikov,


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, respectively.

This paper is dedicated to the memory of V.V.Zhikov

Keywords: Homogenization, Gamma-convergence, thin structures, p-connectedness, networks