The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric
Journal of convex analysis, Tome 22 (2015) no. 1, pp. 19-36
The Γ-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of "two-phase" type, that is, the metric coefficient takes values in {1, β}, with β sufficiently large. The metric coefficient takes the value β on squares, the size of which are controlled by a single parameter. We find a family of examples of limiting Finsler metrics that are piecewise affine with infinitely many lines of discontinuity. Such an example provides insight into how the limit metric behaves under variations of the underlying microscopic Riemannian geometry, with implications for attempts to compute such metrics numerically.
@article{JCA_2015_22_1_JCA_2015_22_1_a1,
author = {H. Schwetlick and D. C. Sutton and J. Zimmer},
title = {The {Finsler} {Metric} {Obtained} as the {\ensuremath{\Gamma}-limit} of a {Generalised} {Manhattan} {Metric}},
journal = {Journal of convex analysis},
pages = {19--36},
year = {2015},
volume = {22},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a1/}
}
TY - JOUR AU - H. Schwetlick AU - D. C. Sutton AU - J. Zimmer TI - The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric JO - Journal of convex analysis PY - 2015 SP - 19 EP - 36 VL - 22 IS - 1 UR - http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a1/ ID - JCA_2015_22_1_JCA_2015_22_1_a1 ER -
%0 Journal Article %A H. Schwetlick %A D. C. Sutton %A J. Zimmer %T The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric %J Journal of convex analysis %D 2015 %P 19-36 %V 22 %N 1 %U http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a1/ %F JCA_2015_22_1_JCA_2015_22_1_a1
H. Schwetlick; D. C. Sutton; J. Zimmer. The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric. Journal of convex analysis, Tome 22 (2015) no. 1, pp. 19-36. http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a1/