Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope

Ambrosio Luigi - Colombo Maria - Di Marino Simone

year: 2012
journal: Advanced Studies in Pure Mathematics
abstract: In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,d,m)$, $1<q<\infty$, in metric measure spaces $(X,d,m)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheeger's work because no Poincar\'e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $m$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

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