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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])
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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

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