Edmunds D.E., Evans W.D. Fractional Sobolev Spaces and Inequalities

Cambridge: Cambridge University Press, 2022. — 169 p.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.Preliminaries
Integration
Banach Spaces
Function Spaces
The Palais–Smale Condition
Classical Sobolev Spaces
Basic Definitions
Fundamental Results
Fractional Sobolev Spaces
Definitions
Basic Properties
An Approach via Interpolation Theory
Connections with the Laplacian
Eigenvalues of the Fractional p-Laplacian
Fundamentals
The Spectrum
Inequalities of Faber–Krahn Type
Classical (Local) Hardy Inequalities
Inequalities on Rn
Geometric Properties of Ω
Convex Domains
The Mean Distance Function
Extensions of Hardy’s Inequality
Discrete Laptev–Weidl Type Inequalities
Fractional Analogues
Special Cases and Consequences
General Domains
Fractional Hardy Inequality with a Remainder Term
Classical and Fractional Inequalities of Rellich Type
The Classical Inequalities
Fractional Rellich Inequalities in Rn
Fractional Rellich Inequalities in General Domains
Higher-Order Fractional Hardy–Rellich Inequalities
Higher-Order Inequality with a Remainder
Higher-Order Classical InequalitiesSymbol Index
Author Index
Subject Index