Skip to main content Accessibility help
×
  1. Home
  2. Books
  3. Fractional Sobolev Spaces and Inequalities
  • Cited by 13
Publisher:
Cambridge University Press
Online publication date:
October 2022
Print publication year:
2022
Online ISBN:
9781009254625
DOI:
https://doi.org/10.1017/9781009254625
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
Series:
Cambridge Tracts in Mathematics (230)
Information

Book description

The fractional Sobolev spaces studied in the book were introduced in the 1950s by Aronszajn, Gagliardo and Slobodeckij in an attempt to fill the gaps between the classical Sobolev spaces. They provide a natural home for solutions of a vast, and rapidly growing, number of questions involving differential equations and non-local effects, ranging from financial modelling to ultra-relativistic quantum mechanics, emphasising the need to be familiar with their fundamental properties and associated techniques. Following an account of the most basic properties of the fractional spaces, two celebrated inequalities, those of Hardy and Rellich, are discussed, first in classical format (for which a survey of the very extensive known results is given), and then in fractional versions. This book will be an Ideal resource for researchers and graduate students working on differential operators and boundary value problems.

Reviews

‘Although introduced in the mid-1950s, the fractional spaces Ws p (Ω) have attracted much recent interest because of their many applications, in probability, continuum mechanics, mathematical biology and elsewhere. This monograph is a concise account of the fractional Banach spaces Ws p (Ω) and their embedding theorems, and of inequalities such as the fractional analogues of those attributed to Hardy and Rellich in the case of Sobolev spaces. While classical and fractional Sobolev spaces have much in common, it is noted that they differ in important respects, and results are derived from the basic definitions, rather than being inferred from the theory of other function spaces with which Ws p (Ω) can be identified, but only when the boundary of Ω is sufficiently regular. The text is a useful guide to the classical and emerging literature.’

John Toland - University of Bath

‘A comprehensive treatise of the theory of fractional Sobolev spaces, defined either on the ambient Euclidean space, or on its generic open subset, and the related inequalities. Striking dissimilarities are discovered in comparison to the classical theory. Contemporary challenging problems are tackled, such as the questions concerning the geometry of underlying domains, the effect of symmetrization techniques, the impact of interpolation, relations to Besov spaces, questions of compactness of embeddings, and more. The book will be of great use to graduate students and researchers of a wide array of scientific interests ranging from partial differential equations and calculus of variations, to approximation and interpolation theory, to the theory of function spaces and related areas. All of the material is accessible through real-variable methods. The only prerequisites, namely basic knowledge of measure theory and Lebesgue integration, will therefore be met by any standard graduate course in real analysis.’

Luboš Pick - Charles University, Prague

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.

Accessibility standard: Unknown

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.