Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces

Abstract

Abstract. This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory study rectifiability issues related to quasiminimal sets in strong A ∞ -weighted Euclidean case.

The quantitative isoperimetric inequality and related topics

“…Therefore it is not surprising that although quasiminimality is a very weak property, yet area quasi-minimizers have some kind of mild regularity. This is indeed the content of the next result which was first proved by David and Semmes in [51] and then extended by Kinnunen et al [88] to the metric spaces setting.…”

A Federer-style characterization of sets of finite perimeter on metric spaces

“…If x ∈ X, 0 < r < R < 1 8 diam X, and A ⊂ B(x, r), then there exists E ⊂ X that is a solution of the K A,0 (B(x, R))-obstacle problem with P (E, X) ≤ cap 1 (A, B(x, R)). The following fact and its proof are similar to [16,Lemma 3.2].…”

Stability and Continuity of Functions of Least Gradient

“…Observe that in [21], this is proved when u is the characteristic function of a set, but the proof works for more general functions as well.…”

“…Proof. According to Theorem 6.1, E u is a set of quasiminimal surface in Ω × R ⊂ X × R equipped with d ∞ and µ × H 1 , and hence by [21] has the properties of [Ω×R]∩∂ * E u = [Ω×R]∩∂ E u and porosity, where E u is defined according to (4.1). Since [Ω × R] ∩ ∂ * E u = [Ω × R] ∩ ∂ E u , for any x ∈ Ω we necessarily have (x, t) ∈ E u for t < u ∧ (x) and (x, t) / ∈ E u for t > u ∨ (x).…”