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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.

Sobolev spaces. --- Inequalities (Mathematics)

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Differential calculus. --- Sobolev spaces. --- Calcul différentiel --- Sobolev, Espaces de --- Differential calculus --- Sobolev spaces --- Calcul différentiel

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The authors consider a parabolic problem with degeneracy in the interior of the spatial domain, and they focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence.

Inequalities (Mathematics) --- Differential equations, Parabolic. --- Carleman theorem. --- Sobolev spaces.

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"La première partie de cet ouvrage, développe des résultats "abstraits" d'analyse fonctionnelle. La seconde partie concerne l'étude d'espaces fonctionnels "concrets" qui interviennent en théorie des équations aux dérivées partielles. Ce livre montre comment des théorèmes d'existence "abstraits" permettent de résoudre des équations aux dérivées partielles."[SUDOC]

Analyse fonctionnelle. --- Sobolev, Espaces de. --- Functional analysis. --- Sobolev spaces. --- Sobolev spaces --- Sobolev, Espaces de --- Functional analysis --- Analyse fonctionnelle

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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.

Metric spaces. --- Sobolev spaces. --- Spaces, Sobolev --- Function spaces --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology