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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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Geology --- Interpolation spaces. --- Spatial analysis (Statistics) --- Analysis, Spatial (Statistics) --- Correlation (Statistics) --- Spatial systems --- Spaces, Interpolation --- Function spaces --- Geological statistics --- Geostatistics --- Statistical methods.

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In this book we suggest a unified method of constructing near-minimizers for certain important functionals arising in approximation, harmonic analysis and ill-posed problems and most widely used in interpolation theory. The constructions are based on far-reaching refinements of the classical Calderón–Zygmund decomposition. These new Calderón–Zygmund decompositions in turn are produced with the help of new covering theorems that combine many remarkable features of classical results established by Besicovitch, Whitney and Wiener. In many cases the minimizers constructed in the book are stable (i.e., remain near-minimizers) under the action of Calderón–Zygmund singular integral operators. The book is divided into two parts. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a self-contained presentation of the main topic. There we discuss the classical covering results mentioned above, various spectacular applications of the classical Calderón–Zygmund decompositions, and the relationship of all this to real interpolation. It also serves as a quick introduction to such important topics as spaces of smooth functions or singular integrals.

Extremal problems (Mathematics). --- Interpolation. --- Matrices. --- Singular integrals. --- Interpolation --- Interpolation spaces --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Interpolation spaces. --- Spaces, Interpolation --- Mathematics. --- Approximation theory. --- Functional analysis. --- Functions of real variables. --- Real Functions. --- Approximations and Expansions. --- Functional Analysis. --- Function spaces --- Approximation theory --- Numerical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Real variables --- Functions of complex variables

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This volume is dedicated to the centenary of the outstanding mathematician of the XXth century Sergey Sobolev and, in a sense, to his celebrated work On a theorem of functional analysis published in 1938, exactly 70 years ago, where the original Sobolev inequality was proved. This double event is a good case to gather experts for presenting the latest results on the study of Sobolev inequalities which play a fundamental role in analysis, the theory of partial differential equations, mathematical physics, and differential geometry. In particular, the following topics are discussed: Sobolev type inequalities on manifolds and metric measure spaces, traces, inequalities with weights, unfamiliar settings of Sobolev type inequalities, Sobolev mappings between manifolds and vector spaces, properties of maximal functions in Sobolev spaces, the sharpness of constants in inequalities, etc. The volume opens with a nice survey reminiscence My Love Affair with the Sobolev Inequality by David R. Adams. Contributors include: David R. Adams (USA); Daniel Aalto (Finland) and Juha Kinnunen (Finland); Sergey Bobkov (USA) and Friedrich Götze (Germany); Andrea Cianchi (Italy); Donatella Danielli (USA), Nicola Garofalo (USA), and Nguyen Cong Phuc (USA); David E. Edmunds (UK) and W. Desmond Evans (UK); Piotr Hajlasz (USA); Vladimir Maz'ya (USA-UK-Sweden) and Tatyana Shaposhnikova USA-Sweden); Luboš Pick (Czech Republic); Yehuda Pinchover (Israel) and Kyril Tintarev (Sweden); Laurent Saloff-Coste (USA); Nageswari Shanmugalingam (USA).

Interpolation spaces. --- Sobolev spaces. --- Sobolev, S. L. --(Sergei? L'vovich), --1908-1989. --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Function spaces. --- Spaces, Function --- Spaces, Sobolev --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Partial differential equations. --- Functions of real variables. --- Numerical analysis. --- Mathematical optimization. --- Analysis. --- Real Functions. --- Partial Differential Equations. --- Functional Analysis. --- Optimization. --- Numerical Analysis. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Real variables --- Functions of complex variables --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.1 Mathematical analysis --- Math --- Science --- Functional analysis --- Function spaces --- Global analysis (Mathematics). --- Differential equations, partial. --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Sobolev spaces --- Interpolation spaces --- Sobolev, S L - (Sergeĭ Lʹvovich), - 1908-1989

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The mathematical works of S.L.Sobolev were strongly motivated by particular problems coming from applications. In his celebrated book Applications of Functional Analysis in Mathematical Physics, 1950 and other works, S.Sobolev introduced general methods that turned out to be very influential in the study of mathematical physics in the second half of the XXth century. This volume, dedicated to the centenary of S.L. Sobolev, presents the latest results on some important problems of mathematical physics describing, in particular, phenomena of superconductivity with random fluctuations, wave propagation, perforated domains and bodies with defects of different types, spectral asymptotics for Dirac energy, Lam'e system with residual stress, optimal control problems for partial differential equations and inverse problems admitting numerous interpretations. Methods of modern functional analysis are essentially used in the investigation of these problems. Contributors include: Mikhail Belishev (Russia); Andrei Fursikov (Russia), Max Gunzburger (USA), and Janet Peterson (USA); Victor Isakov (USA) and Nanhee Kim (USA); Victor Ivrii (Canada); Irena Lasiecka (USA) and Roberto Triggiani (USA); Vladimir Maz'ya (USA-UK-Sweden) and Alexander Movchan (UK); Michael Taylor (USA).

Interpolation spaces. --- Sobolev spaces. --- Sobolev, S. L. --(Sergei? L'vovich), --1908-1989. --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Calculus --- Applied Mathematics --- Function spaces. --- Spaces, Function --- Spaces, Sobolev --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Partial differential equations. --- Functions of real variables. --- Numerical analysis. --- Mathematical optimization. --- Analysis. --- Real Functions. --- Partial Differential Equations. --- Functional Analysis. --- Optimization. --- Numerical Analysis. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Real variables --- Functions of complex variables --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.1 Mathematical analysis --- Math --- Science --- Functional analysis --- Function spaces --- Global analysis (Mathematics). --- Differential equations, partial. --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Sobolev spaces --- Interpolation spaces --- Sobolev, S L - (Sergeĭ Lʹvovich), - 1908-1989