Sobolev Spaces on Metric Measure Spaces (eBook)

An Approach Based on Upper Gradients

Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson (Autoren)

eBook Download: PDF

2015
Cambridge University Press (Verlag)
978-1-316-23725-0 (ISBN)

Infos

Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger''s stability theorem for Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.

Produkt-Details

• Erscheint lt. Verlag: 5.2.2015
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-10: 1-316-23725-7 / 1316237257
• ISBN-13: 978-1-316-23725-0 / 9781316237250