## Advanced Studies in Pure Mathematics

### 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.

#### Article information

Dates
Revised: 25 May 2014
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/ euclid.aspm/1539916032

Digital Object Identifier
doi:10.2969/aspm/06710001

Mathematical Reviews number (MathSciNet)
MR3587446

Zentralblatt MATH identifier
1370.46018

#### Citation

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. https://projecteuclid.org/euclid.aspm/1539916032