Journal of Convex Analysis 22 (2015), No. 1, 019--036
Copyright Heldermann Verlag 2015
The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric
Hartmut Schwetlick
Dept. of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
schwetlick@maths.bath.ac.uk
Daniel C. Sutton
Dept. of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
d.sutton@bath.ac.uk
Johannes Zimmer
Dept. of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
zimmer@maths.bath.ac.uk
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.
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