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Fractional Sobolev Spaces and Inequalities

Provides an account of fractional Sobolev spaces emphasising applications to famous inequalities. Ideal for graduates and researchers.

D. E. Edmunds (Author), W. D. Evans (Author)

9781009254632, Cambridge University Press

Hardback, published 13 October 2022

175 pages
23.5 x 15.8 x 1.5 cm, 0.4 kg

'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

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.

1. Preliminaries
2. Classical Sobolev spaces
3. Fractional Sobolev spaces
4. Eigenvalues of the fractional p-Laplacian
5. Classical (local) Hardy inequalities
6. Fractional analogues
7. Classical and fractional inequalities of Rellich type
References
Symbol index
Author index
Index of terms.

Subject Areas: Differential calculus & equations [PBKJ], Functional analysis & transforms [PBKF]

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