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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

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Sobolev spaces were firstly defined by the Russian mathematician, Sergei L. Sobolev (1908-1989) in the 1930s. Several properties of these spaces have been studied by mathematicians until today. Functions that account for existence and uniqueness, asymptotic behavior, blow up, stability and instability of the solution of many differential equations that occur in applied and in engineering sciences are carried out with the help of Sobolev spaces and embedding theorems in these spaces. An Introduction to Sobolev Spaces provides a brief introduction to Sobolev spaces at a simple level with illustrated examples. Readers will learn about the properties of these types of vector spaces and gain an understanding of advanced differential calculus and partial difference equations that are related to this topic. The contents of the book are suitable for undergraduate and graduate students, mathematicians, and engineers who have an interest in getting a quick, but carefully presented, mathematically sound, basic knowledge about Sobolev Spaces.

Sobolev spaces. --- Interpolation spaces. --- Differential equations, Partial. --- Partial differential equations --- Spaces, Interpolation --- Function spaces --- Spaces, Sobolev

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Mathematical analysis --- Multipliers (Mathematical analysis) --- Sobolev spaces --- Spaces, Sobolev --- Function spaces --- Functional analysis --- Harmonic analysis --- Sobolev spaces. --- Sobolev, Espaces de. --- Multiplicateurs (analyse mathématique)

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Espais de Sobolev --- Espais funcionals --- Manifolds (Mathematics) --- Sobolev spaces. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Spaces, Sobolev --- Function spaces --- Geometry, Differential --- Topology

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Analytical spaces --- Espaces de Sobolev --- Menigvuldigheden van Riemann --- Riemannian manifolds --- Ruimten van Sobolev --- Sobolev [Espaces de ] --- Sobolev [Ruimten van ] --- Sobolev spaces --- Spaces [Sobolev ] --- Variétés de Riemann --- Sobolev spaces. --- Riemannian manifolds. --- Periodicals