Spectral Geometry of Partial Differential Operators
The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry...
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| Médium: | E-kniha Kniha |
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Boca Raton
CRC Press
2020
No Funder Information Available Taylor & Francis Chapman & Hall |
| Vydání: | 1 |
| Edice: | Chapman & Hall/CRC Monographs and Research Notes in Mathematics |
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| ISBN: | 1138360716, 9781138360716, 9780429432965, 9780429780578, 0429780567, 9780429780554, 0429780559, 9780429780561, 0429432968, 0429780575 |
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| Abstract | The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory. |
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| AbstractList | The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations.Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features:Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operatorsAimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciencesProvides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods.Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory. The present is an attempt to collect a number of properties emerging in the recent research describing certain features of the theory of partial differential equations that can be attributed to the field of spectral geometry. Both being vast fields, our attempt is not to give a comprehensive account of the whole theory but to provide the reader with a quick introduction to a number of its important aspects. The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory. 1. Function spaces. 2. Foundations of linear operator theory. 3. Elements of the spectral theory of differential operators. 4. Symmetric decreasing rearrangements and applications. 5. Inequalities of spectral geometry. Michael Ruzhansky is a Senior Full Professor of Mathematics at Ghent University, Belgium, and a Professor of Mathematics at the Queen Mary University of London, United Kindgdom. He is currently also an Honorary Professor of Pure Mathematics at Imperial College London, where he has been working in the period 2000-2018. His research is devoted to different topics in the analysis of partial differential equations, harmonic and non-harmonic analysis, spectral theory, microlocal analysis, as well as the operator theory and functional inequalities on groups. His research was recognised by the ISAAC Award 2007, Daiwa Adrian Prize 2010, as well as by the Ferran Sunyer I Balaguer Prizes in 2014 and 2018. Makhmud Sadybekov is a Kazakhstani mathematician who graduated from the Kazakh State University (Almaty, Kazakhstan) in 1985 and received his doctorate in physical-mathematical sciences in 1993. He is a specialist in the field of Ordinary Differential Equations, Partial Differential Equations, Equations of Mathematical Physics, Functional Analysis, Operators Theory. Currently he is Director General at the Institute of Mathematics and Mathematical Modeling in Almaty, Kazakhstan. Durvudkhan Suragan an associate professor at Nazarbayev University. He won the Ferran Sunyer i Balaguer Prize in 2018. He has previously worked in spectral geometry, and in the theory of subelliptic inequalities at Imperial College London as a research associate and as a leading researcher in the Institute of Mathematics and Mathematical Modeling. Open access – no commercial reuse The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory. |
| Author | Suragan, Durvudkhan Ruzhansky, Michael Sadybekov, Makhmud |
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| Keywords | Dirichlet Laplacian Gaseous Stars Hardy Littlewood Inequality Smooth Nonnegative Function Vlasov Poisson Equations Euler Poisson System Cauchy Sequence Sobolev Space Ordinary Differential Equation Nonnegative Measurable Functions Vlasov Poisson System Generalised Derivative Separable Infinite Dimensional Hilbert Space Banach Space Galactic Dynamics Symmetric Rearrangement Linear Normed Space Linear Space Symmetric Steady States Lebesgue Integral Online Lecture Note Hilbert Space |
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| Notes | A Chapman & Hall book Includes bibliographical references (p. 353-362) and index |
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| Snippet | The present is an attempt to collect a number of properties emerging in the recent research describing certain features of the theory of partial differential... The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral... |
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| SubjectTerms | Advanced Mathematics Advanced Topics Analysis - Mathematics Applied mathematics Banach Space bounded linear operators Calculus and mathematical analysis Cauchy Sequence Differential calculus and equations Dirichlet Laplacian Euler Poisson System Fredholm operators Functional Analysis Functional analysis and transforms Generalised Derivative Hardy Littlewood Inequality Hilbert Space Lebesgue integral linear differential operators Linear Normed Space Linear Space Mathematical Analysis Mathematical Physics Mathematics Mathematics and Science MATHnetBASE Nonnegative Measurable Functions Operator Theory Partial differential operators Physics Probability and statistics Pure Mathematics Riesz' inequality SCI-TECHnetBASE Separable Infinite Dimensional Hilbert Space Spectral geometry spectral invariants STMnetBASE Symmetric Rearrangement Vlasov Poisson Equations Vlasov Poisson System |
| Title | Spectral Geometry of Partial Differential Operators |
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