The triangle inequalities and lower semi-continuity of surface energy of partitions
The triangle inequalities and lower semi-continuity of surface energy of partitions
复制标题
三角不等式与隔壁表面能的下半连续性
DOI：
10.1017/s0308210506000837
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
David G. Caraballo
中科院分区：
文献类型：
--
作者：
David G. Caraballo
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.
