Abstract

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.

Original languageEnglish
Pages (from-to)19-36
Number of pages18
JournalJournal of Convex Analysis
Volume22
Issue number1
Publication statusPublished - 1 Jan 2015

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

Cite this

The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric. / Schwetlick, Hartmut; Sutton, Daniel C.; Zimmer, Johannes.

In: Journal of Convex Analysis, Vol. 22, No. 1, 01.01.2015, p. 19-36.

Research output: Contribution to journalArticle

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AB - 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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