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Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential geometry etc. The theory of these spaces is of interest in itself being a beautiful domain of mathematics. The present volume includes basics on Sobolev spaces, approximation and extension theorems, embedding and compactness theorems, their relations with isoperimetric and isocapacitary inequalities, capacities with applications to spectral theory of elliptic differential operators as well as pointwise inequalities for derivatives. The selection of topics is mainly influenced by the author’s involvement in their study, a considerable part of the text is a report on his work in the field. Part of this volume first appeared in German as three booklets of Teubner-Texte zur Mathematik (1979,1980). In the Springer volume “Sobolev Spaces”, published in English in 1985, the material was expanded and revised. The present 2nd edition is enhanced by many recent results and it includes new applications to linear and nonlinear partial differential equations. New historical comments, five new chapters and a significantly augmented list of references aim to create a broader and modern view of the area.

Electronic books. -- local. --- Functional analysis. --- Sobolev spaces. --- Sobolev spaces --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Functional calculus --- Spaces, Sobolev --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Analysis. --- Calculus of variations --- Functional equations --- Integral equations --- Function spaces --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis

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Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are in the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integrability of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc. Two short biographical articles on the works of Sobolev in the 1930's and foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included. Contributors include: Vasilii Babich (Russia); Yuri Reshetnyak (Russia); Hiroaki Aikawa (Japan); Yuri Brudnyi (Israel); Victor Burenkov (Italy) and Pier Domenico Lamberti (Italy); Serban Costea (Canada) and Vladimir Maz'ya (USA-UK-Sweden); Stephan Dahlke (Germany) and Winfried Sickel (Germany); Victor Galaktionov (UK), Enzo Mitidieri (Italy), and Stanislav Pokhozhaev (Russia); Vladimir Gol'dshtein (Israel) and Marc Troyanov (Switzerland); Alexander Grigor'yan (Germany) and Elton Hsu (USA); Tunde Jakab (USA), Irina Mitrea (USA), and Marius Mitrea (USA); Sergey Nazarov (Russia); Grigori Rozenblum (Sweden) and Michael Solomyak (Israel); Hans Triebel (Germany).

Interpolation spaces. --- Sobolev spaces. --- Sobolev, S. L. (Sergei Lvovich), 1908. --- Calculus --- Applied Mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Differential equations, Partial. --- Functional analysis. --- Functional calculus --- Partial differential equations --- Spaces, Sobolev --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- Numerical analysis. --- Mathematical optimization. --- Physics. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Partial Differential Equations. --- Functional Analysis. --- Optimization. --- Numerical Analysis. --- Calculus of variations --- Functional equations --- Integral equations --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- 517.1 Mathematical analysis --- Math --- Science --- Function spaces --- Global analysis (Mathematics). --- Differential equations, partial. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical physics. --- Physical mathematics --- Physics --- Sobolev spaces --- Interpolation spaces --- Sobolev, S L - (Sergeĭ Lʹvovich), - 1908-1989

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Vladimir Maz'ya (born 1937) is an outstanding mathematician who systematically made fundamental contributions to a wide array of areas in mathematical analysis and in the theory of partial differential equations. In this fascinating book he describes the first thirty years of his life in Leningrad (now St. Petersburg). He starts with the story of his family, speaks about his childhood, the high school and university years, and recalls his formative years as a mathematician. Behind the author's personal recollections, with his own joys, sorrows and hopes, one sees a vivid picture of those times in the former Sovjet Union. He speaks warmly about his friends, both outside and inside the world of mathematics, about discovering his passion for mathematics and his early achievements, and about a number of mathematicians who influenced his professional life. The book is written in a highly readable and inviting style, spiced with the occasional touch of humor.

Mathematics --- wiskunde

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Vladimir Maz'ya (born 1937) is an outstanding mathematician who systematically made fundamental contributions to a wide array of areas in mathematical analysis and in the theory of partial differential equations. In this fascinating book he describes the first thirty years of his life in Leningrad (now St. Petersburg). He starts with the story of his family, speaks about his childhood, the high school and university years, and recalls his formative years as a mathematician. Behind the author's personal recollections, with his own joys, sorrows and hopes, one sees a vivid picture of those times in the former Sovjet Union. He speaks warmly about his friends, both outside and inside the world of mathematics, about discovering his passion for mathematics and his early achievements, and about a number of mathematicians who influenced his professional life. The book is written in a highly readable and inviting style, spiced with the occasional touch of humor.

Mathematicians --- Mathematics. --- History of Mathematical Sciences. --- Mathematics, general. --- Math --- Science --- History. --- Annals --- Auxiliary sciences of history --- Mazʹi︠a︡, V. G. --- Мазья, В. Г. --- Mazʹya, V. G. --- Mazja, W. G. --- Mazʹja, V. G. --- Mazʹi︠a︡, Vladimir Gilelevich --- Мазья, Владимир Гилелевич --- Mazʹya, Vladimir --- Mazʹya, V.

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Functional analysis --- Differential equations --- differentiaalvergelijkingen --- Laplacetransformatie --- functies (wiskunde)