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Fractional Sobolev Spaces and Inequalities

2022-10-13 Mathematics
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Release : 2022-10-13
Genre : Mathematics
Kind : eBook
Book Rating : 642/5 ( reviews)

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Book Synopsis Fractional Sobolev Spaces and Inequalities by : D. E. Edmunds

Download or read book Fractional Sobolev Spaces and Inequalities written by D. E. Edmunds. This book was released on 2022-10-13. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A First Course in Fractional Sobolev Spaces

2023-03-17 Mathematics
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Release : 2023-03-17
Genre : Mathematics
Kind : eBook
Book Rating : 538/5 ( reviews)

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Book Synopsis A First Course in Fractional Sobolev Spaces by : Giovanni Leoni

Download or read book A First Course in Fractional Sobolev Spaces written by Giovanni Leoni. This book was released on 2023-03-17. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a gentle introduction to fractional Sobolev spaces which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagliardo-Nirenberg type interpolation inequalities, and trace theory. The third part explores some applications: interior regularity for the Poisson problem with the right-hand side in a fractional Sobolev space and some basic properties of the fractional Laplacian. The first part of the book is accessible to advanced undergraduates with a strong background in integration theory; the second part, to graduate students having familiarity with measure and integration and some functional analysis. Basic knowledge of Sobolev spaces would help, but is not necessary. The book can also serve as a reference for mathematicians working in the calculus of variations and partial differential equations as well as for researchers in other disciplines with a solid mathematics background. It contains several exercises and is self-contained.

Fractional Sobolev Spaces and Inequalities

2022-10-31 Mathematics
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Release : 2022-10-31
Genre : Mathematics
Kind : eBook
Book Rating : 634/5 ( reviews)

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Book Synopsis Fractional Sobolev Spaces and Inequalities by : D. E. Edmunds

Download or read book Fractional Sobolev Spaces and Inequalities written by D. E. Edmunds. This book was released on 2022-10-31. Available in PDF, EPUB and Kindle. Book excerpt: Provides an account of fractional Sobolev spaces emphasising applications to famous inequalities. Ideal for graduates and researchers.

Around the Research of Vladimir Maz'ya I

2009-12-02 Mathematics
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Release : 2009-12-02
Genre : Mathematics
Kind : eBook
Book Rating : 416/5 ( reviews)

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Book Synopsis Around the Research of Vladimir Maz'ya I by : Ari Laptev

Download or read book Around the Research of Vladimir Maz'ya I written by Ari Laptev. This book was released on 2009-12-02. Available in PDF, EPUB and Kindle. Book excerpt: The fundamental contributions of Professor Maz'ya to the theory of function spaces and especially Sobolev spaces are well known and often play a key role in the study of different aspects of the theory, which is demonstrated, in particular, by presented new results and reviews from world-recognized specialists. Sobolev type spaces, extensions, capacities, Sobolev inequalities, pseudo-Poincare inequalities, optimal Hardy-Sobolev-Maz'ya inequalities, Maz'ya's isocapacitary inequalities in a measure-metric space setting and many other actual topics are discussed.

Fractional Sobolev Inequalities

2014 Inequalities (Mathematics)
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Release : 2014
Genre : Inequalities (Mathematics)
Kind : eBook
Book Rating : 964/5 ( reviews)

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Book Synopsis Fractional Sobolev Inequalities by : Joaquim Martín

Download or read book Fractional Sobolev Inequalities written by Joaquim Martín. This book was released on 2014. Available in PDF, EPUB and Kindle. Book excerpt: The authors obtain new oscillation inequalities in metric spaces in terms of the Peetre $K$-functional and the isoperimetric profile. Applications provided include a detailed study of fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular, the authors include a detailed study of Gaussian measures as well as probability measures between Gaussian and exponential. They show a kind of reverse Polya-Szego principle that allows them to obtain continuity as a self-improvement from boundedness, using symmetrization inequalities. The authors' methods also allow for precise estimates of growth envelopes of generalized Sobolev and Besov spaces on metric spaces. The authors also consider embeddings into BMO and their connection to Sobolev embeddings.

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