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arXiv · 1212.3779

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

Luigi Ambrosio·Maria Colombo·Simone Di Marino
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Abstract

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $Γ$-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 $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

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Luigi Ambrosio, Maria Colombo, Simone Di Marino. 2012-12-16. Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. https://arxiv.org/abs/1212.3779

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