Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T20:40:45.926Z Has data issue: false hasContentIssue false
  • English
  • Français

The triangle inequalities and lower semi-continuity of surface energy of partitions

Published online by Cambridge University Press:  26 May 2009

David G. Caraballo
David G. Caraballo
Affiliation:
Department of Mathematics, St Mary's Hall, Georgetown University, Washington, DC 20057-1233, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the triangle inequalities in general are not equivalent to lower semi-continuity of surface energy. In 1990, Ambrosio and Braides proved that the two conditions are equivalent when the number of regions, s, is three. They gave an example in the plane showing that the triangle inequalities do not imply lower semi-continuity when s ≥ 6. The cases when s = 4 and s = 5 have remained open. In this paper, we resolve these open questions and show that, in ℝm, the triangle inequalities are sufficient for lower semi-continuity if and only if s = 3.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 3 , June 2009 , pp. 449 - 457
Copyright
Copyright © Royal Society of Edinburgh 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)