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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

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

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

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1539916032

Digital Object Identifier
doi:10.2969/aspm/06710001

Mathematical Reviews number (MathSciNet)
MR3587446

Zentralblatt MATH identifier
1370.46018

Subjects
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

Keywords
Sobolev spaces metric measure spaces weak gradients

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


Export citation