Advanced Studies in Pure Mathematics

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

Luigi Ambrosio, Maria Colombo, and Simone Di Marino

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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.

Article information

Variational Methods for Evolving Objects, L. Ambrosio, Y. Giga, P. Rybka and Y. Tonegawa, eds. (Tokyo: Mathematical Society of Japan, 2015), 1-58

Received: 27 December 2012
Revised: 25 May 2014
First available in Project Euclid: 19 October 2018

Permanent link to this document euclid.aspm/1539916032

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 49M25: Discrete approximations 49Q20: Variational problems in a geometric measure-theoretic setting 58J35: Heat and other parabolic equation methods 35K90: Abstract parabolic equations 31C25: Dirichlet spaces

Sobolev spaces metric measure spaces weak gradients


Ambrosio, Luigi; Colombo, Maria; Marino, Simone Di. Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. Variational Methods for Evolving Objects, 1--58, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06710001.

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