Issue |
ESAIM: COCV
Volume 23, Number 3, July-September 2017
|
|
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Page(s) | 889 - 912 | |
DOI | https://doi.org/10.1051/cocv/2016018 | |
Published online | 29 March 2017 |
Homogenization of metrics in oscillating manifolds∗
1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della ricerca scientifica 1, 00133 Roma, Italy.
braides@mat.uniroma2.it
2 Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Received: 6 November 2015
Revised: 29 January 2016
Accepted: 25 March 2016
We consider energies defined as the Dirichlet integral of curves taking values in fast-oscillating manifolds converging to a linear subspace. We model such manifolds as subsets of Rm + m′ described by a constraint (xm + 1,...,xm′) = δϕ(x1/ε,...,xm/ε) where ε is the period of the oscillation, δ its amplitude and ϕ its profile. The interesting case is ε<<δ<< 1, in which the limit of the energies is described by a Finsler metric on Rm which is defined by optimizing the contribution of oscillations on each level set {ϕ = c}. The formulas describing the limit mix homogenization and convexification processes, highlighting a multi-scale behaviour of optimal sequences. We apply these formulas to show that we may obtain all (homogeneous) symmetric Finsler metrics larger than the Euclidean metric as limits in the case of oscillating surfaces in R3.
Mathematics Subject Classification: 35B27 / 49J45 / 58B20
Key words: Homogenization / oscillating manifolds / Finsler metrics
© EDP Sciences, SMAI 2017
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