Article ID Journal Published Year Pages File Type
839216 Nonlinear Analysis: Theory, Methods & Applications 2016 44 Pages PDF
Abstract

Let (X,dX,μ)(X,dX,μ) be a metric measure space where XX is locally compact and separable and μμ is a Borel regular measure such that 0<μ(B(x,r))<∞0<μ(B(x,r))<∞ for every ball B(x,r)B(x,r) with center x∈Xx∈X and radius r>0r>0. We define XX to be the set of all positive, finite non-zero regular Borel measures with compact support in XX which are dominated by μμ, and M=X∪{0}M=X∪{0}. By introducing a kind of mass transport metric dMdM on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F:X→RF:X→R, and then for functions f:X→[−∞,∞]f:X→[−∞,∞] by identifying them with the unique element Ff:X→RFf:X→R defined by the mean-value integral:Ff(η)=1‖η‖∫fdη. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space RnRn with Lebesgue measure.

Related Topics
Physical Sciences and Engineering Engineering Engineering (General)
Authors
,