Calculus of Variations and Geometric Measure Theory

L. Ambrosio - M. Colombo - S. Di Marino

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

created by ambrosio on 13 Dec 2012
modified by dimarino on 24 Jul 2018


Accepted Paper

Inserted: 13 dec 2012
Last Updated: 24 jul 2018

Journal: Advanced Studies in Pure Mathematics: "Variational methods for evolving objects"
Volume: 67
Pages: 1-58
Year: 2012

ArXiv: 1212.3779 PDF


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 $\Gamma$-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 $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

Tags: GeMeThNES
Keywords: Sobolev spaces, Metric measure spaces, Weak Gradients