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E-grāmata: Spectral Geometry of Partial Differential Operators

, (Nazarbayev University, Khazakhstan), (Ghent University, Belgium)
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Spectral Geometry of Partial Differential OperatorsPalielināt
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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.
Preface ix
Authors xi
1 Functional spaces
1(24)
1.1 Normed spaces
1(4)
1.2 Hilbert spaces
5(1)
1.3 Examples of basic functional spaces
6(1)
1.4 The concept of Lebesgue integral
7(1)
1.5 Lebesgue spaces
8(3)
1.6 Sobolev spaces
11(4)
1.7 Subspaces
15(3)
1.8 Initial concept of embedding of spaces
18(2)
1.9 Separable spaces
20(5)
2 Foundations of the linear operator theory
25(90)
2.1 Definition of operator
25(1)
2.2 Linear operators
26(2)
2.3 Linear bounded operators
28(10)
2.4 Space of bounded linear operators
38(1)
2.5 Extension of a linear operator with respect to continuity
39(3)
2.6 Linear functionals
42(3)
2.7 Inverse operators
45(3)
2.8 Examples of invertible operators
48(5)
2.9 The contraction mapping principle
53(3)
2.10 Normally solvable operators
56(3)
2.11 Restrictions and extensions of linear operators
59(2)
2.12 Closed operators
61(8)
2.13 Closure of differential operators in L2 (a, b)
69(3)
2.14 General concept of strong solutions of the problem
72(2)
2.15 Compact operators
74(6)
2.16 Volterra operators
80(4)
2.17 Structure of the dual space
84(1)
2.18 Adjoint to a bounded operator
85(6)
2.19 Adjoint to unbounded operators
91(14)
2.20 Two examples of nonclassical problems
105(10)
3 Elements of the spectral theory of differential operators
115(134)
3.1 Spectrum of finite-dimensional operators
115(6)
3.2 The resolvent and spectrum of an operator
121(6)
3.3 Spectral properties of bounded operators
127(2)
3.4 Spectrum of compact operators
129(10)
3.5 Hilbert-Schmidt theorem and its application
139(3)
3.6 Spectral properties of unbounded operators
142(8)
3.7 Some ordinary differential operators and their spectrum
150(7)
3.8 Spectral theory of the Sturm-Liouville operator
157(11)
3.9 Spectral trace and Hilbert-Schmidt operators
168(9)
3.10 Schatten-von Neumann classes
177(10)
3.11 Regularised trace for a differential operator
187(5)
3.12 Eigenvalues of non-self-adjoint ordinary differential operators of the second order
192(11)
3.13 Biorthogonal systems in Hilbert spaces
203(4)
3.14 Biorthogonal expansions and Riesz bases
207(14)
3.15 Convolutions in Hilbert spaces
221(10)
3.16 Root functions of second-order non-self-adjoint ordinary differential operators
231(18)
4 Symmetric decreasing rearrangements and applications
249(30)
4.1 Symmetric decreasing rearrangements
250(4)
4.1.1 Definitions and examples
250(2)
4.1.2 Some properties
252(2)
4.2 Basic inequalities
254(7)
4.2.1 Hardy-Littlewood inequality
255(1)
4.2.2 Riesz inequality
255(2)
4.2.3 Polya-Szego inequality
257(1)
4.2.4 Talenti's comparison principles
258(3)
4.3 Properties of symmetric rearrangement sequences
261(10)
4.3.1 Burchard-Guo theorem
261(8)
4.3.2 Proof of Burchard-Guo theorem
269(2)
4.4 Applications in mathematical physics
271(8)
4.4.1 Brownian motion in a ball
271(1)
4.4.2 Optimal membrane shape for the deepest bass note
272(1)
4.4.3 Maximiser body of the gravitational field energy
273(1)
4.4.4 Dynamical stability problem of gaseous stars
274(2)
4.4.5 Stability of symmetric steady states in galactic dynamics
276(3)
5 Inequalities of spectral geometry
279(74)
5.1 Introduction
279(3)
5.2 Logarithmic potential operator
282(12)
5.2.1 Spectral geometric inequalities and examples
283(6)
5.2.2 Isoperimetric inequalities over polygons
289(5)
5.3 Riesz potential operators
294(10)
5.3.1 Spectral properties of Rα,Ω
295(4)
5.3.2 Spectral geometric inequalities for Rα,Ω
299(5)
5.4 Bessel potential operators
304(7)
5.4.1 Spectral properties of Bα,Ω
305(1)
5.4.2 Boundary properties of Bα,Ω
306(5)
5.5 Riesz transforms in spherical and hyperbolic geometries
311(8)
5.5.1 Geometric inequalities for the first eigenvalue
313(2)
5.5.2 Geometric inequalities for the second eigenvalue
315(4)
5.6 Heat potential operators
319(10)
5.6.1 Basic properties
320(2)
5.6.2 Spectral geometric inequalities for the heat potential
322(5)
5.6.3 The case of triangular prisms
327(2)
5.7 Cauchy-Dirichlet heat operator
329(10)
5.7.1 Spectral geometric inequalities for the Cauchy-Dirichlet heat operator
329(7)
5.7.2 The case of polygonal cylinders
336(3)
5.8 Cauchy-Robin heat operator
339(4)
5.8.1 Isoperimetric inequalities for the first s-number
339(3)
5.8.2 Isoperimetric inequalities for the second s-number
342(1)
5.9 Cauchy-Neumann and Cauchy-Dirichlet-Neumann heat operators
343(10)
5.9.1 Basic properties
344(2)
5.9.2 On the Szego-Weinberger type inequality
346(2)
5.9.3 Inequalities for the Cauchy-Dirichlet-Neumann operator
348(5)
Bibliography 353(10)
Index 363
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.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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