Homogenization of networks in domains with oscillating boundaries

ABSTRACT

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 $\mathrm{\Gamma }$-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.

KEYWORDS:

• Networks
• homogenization
• thin structures
• p-connectedness
• $\mathrm{\Gamma }$-convergence

AMS SUBJECT CLASSIFICATIONS:

• 49J45
• 35B27
• 74K35

Notes

No potential conflict of interest was reported by the authors.

In memoriam V.V. Zhikov.