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 described by a constraint where is the period of the oscillation, its amplitude and its profile. The interesting case is , in which the limit of the energies is described by a Finsler metric on which is defined by optimizing the contribution of oscillations on each level set . 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 .
Keywords: Homogenization, oscillating manifolds, Finsler metrics
@article{COCV_2017__23_3_889_0, author = {Braides, Andrea and Cancedda, Andrea and Piat, Valeria Chiad\`o}, title = {Homogenization of metrics in oscillating manifolds}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {889--912}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016018}, mrnumber = {3660453}, zbl = {1367.35023}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016018/} }
TY - JOUR AU - Braides, Andrea AU - Cancedda, Andrea AU - Piat, Valeria Chiadò TI - Homogenization of metrics in oscillating manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 889 EP - 912 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016018/ DO - 10.1051/cocv/2016018 LA - en ID - COCV_2017__23_3_889_0 ER -
%0 Journal Article %A Braides, Andrea %A Cancedda, Andrea %A Piat, Valeria Chiadò %T Homogenization of metrics in oscillating manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 889-912 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016018/ %R 10.1051/cocv/2016018 %G en %F COCV_2017__23_3_889_0
Braides, Andrea; Cancedda, Andrea; Piat, Valeria Chiadò. Homogenization of metrics in oscillating manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 889-912. doi : 10.1051/cocv/2016018. http://archive.numdam.org/articles/10.1051/cocv/2016018/
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