Uniform local energy estimates and partial regularity for space partitions

Abstract

For partitions of space into sets of finite perimeter, and for general convex surface energy density functions, we show that surface energy may be locally linearly approximated, with uniform error bounds which hold even when the interfaces and surface energy density functions are non-smooth. We show that locally simple partitions – those satisfying a quite general density ratio condition – have boundaries which locally almost everywhere consist of a single interface, and have boundary closures which are (n − 1) dimensional. Local simplicity typically holds for variational problems in which surface area, surface energy or a surface plus bulk energy sum are to be minimized. Local simplicity rules out pathological boundary behaviour, helps establish lower semicontinuity, existence and regularity results and facilitates the use of finite perimeter partitions in applications in materials science, image processing and other fields.

Keywords:

• partitions
• polycrystals
• clusters
• finite perimeter
• surface energy
• Lebesgue point
• locally simple
• regularity
• covering
• decomposition

AMS Subject Classifications::

• 49Q15
• 52A41
• 49N60

Notes

Note

1. This differs from the terminology used in 17, where such sets are referred to as ℋ ^m measurable and (ℋ ^m , m) rectifiable.