Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces

This paper studies regularity of perimeter 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 fin...

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Bibliographic Details
Published in:The Journal of geometric analysis 2013-10, Vol.23 (4), p.1607-1640
Main Authors: Kinnunen, Juha, Korte, Riikka, Lorent, Andrew, Shanmugalingam, Nageswari
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
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Summary:This paper studies regularity of perimeter 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 to study rectifiability issues related to quasiminimal sets in the strong A ∞ -weighted Euclidean case.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-012-9299-z