6533b837fe1ef96bd12a3399
RESEARCH PRODUCT
The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces
Juha KinnunenHeli TuominenNageswari ShanmugalingamRiikka Kortesubject
Pure mathematicsMathematics(all)General MathematicsApplied Mathematics010102 general mathematicsMathematical analysisBoxing inequalityCaccioppoli setDiscrete measureσ-finite measure01 natural sciencesRelaxed problemCapacitiesTransverse measure0103 physical sciencesComplex measureOuter measureHausdorff measure010307 mathematical physics0101 mathematicsBorel measureFunctions of bounded variationMathematicsdescription
Abstract We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2010-06-01 | Journal de Mathématiques Pures et Appliquées |