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

Abstract

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X, \mathsf{d}, \mathfrak{m})$, $1 \lt q \lt \infty$, in metric measure spaces $(X, \mathsf{d}, \mathfrak{m})$. 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 Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mathfrak{m}$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.