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VOL. 67 | 2015 Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
Luigi Ambrosio, Maria Colombo, Simone Di Marino

Editor(s) Luigi Ambrosio, Yoshikazu Giga, Piotr Rybka, Yoshihiro Tonegawa


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.


Published: 1 January 2015
First available in Project Euclid: 19 October 2018

zbMATH: 1370.46018
MathSciNet: MR3587446

Digital Object Identifier: 10.2969/aspm/06710001

Primary: 31C25, 35K90, 49J52, 49M25, 49Q20, 58J35

Rights: Copyright © 2015 Mathematical Society of Japan


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