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El. knyga: Potential Theory

4.50/5 (2 ratings by Goodreads)
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Apr-2014
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781447164227
Kitos knygos pagal šią temą:
Potential TheoryDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Apr-2014
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781447164227
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Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region.

The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion.

In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics and engineering.



This book presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible.
1 Preliminaries
1(8)
1.1 Notation
1(3)
1.2 Useful Theorems
4(2)
1.3 Force Fields
6(3)
1.3.1 Exercises for Sect. 1.3
7(1)
References
8(1)
2 Laplace's Equation
9(54)
2.1 Introduction
9(1)
2.1.1 Exercises for Sect. 2.1
10(1)
2.2 Green's Theorem
10(1)
2.3 Fundamental Harmonic Function
11(2)
2.3.1 Exercises for Sect. 2.3
13(1)
2.4 The Mean Value Property
13(4)
2.4.1 Exercises for Sect. 2.4
17(1)
2.5 Poisson Integral Formula
17(8)
2.5.1 Exercises for Sect. 2.5
24(1)
2.6 Gauss' Averaging Principle
25(4)
2.6.1 Exercises for Sect. 2.6
28(1)
2.7 The Dirichlet Problem for a Ball
29(9)
2.7.1 Exercises for Sect. 2.7
37(1)
2.8 Kelvin Transformation
38(3)
2.8.1 Exercises for Sect. 2.8
40(1)
2.9 Poisson Integral for Half-Space
41(6)
2.9.1 Exercises for Sect. 2.9
46(1)
2.10 Neumann Problem for a Disk
47(4)
2.10.1 Exercises for Sect. 2.10
50(1)
2.11 Neumann Problem for the Ball
51(6)
2.12 Spherical Harmonics
57(6)
2.12.1 Exercises for Sect. 2.12
61(1)
References
61(2)
3 The Dirichlet Problem
63(58)
3.1 Introduction
63(1)
3.2 Sequences of Harmonic Functions
64(5)
3.2.1 Exercises for Sect. 3.2
69(1)
3.3 Superharmonic Functions
69(6)
3.3.1 Exercises for Sect. 3.3
75(1)
3.4 Properties of Superharmonic Functions
75(6)
3.4.1 Exercises for Sect. 3.4
81(1)
3.5 Approximation of Superharmonic Functions
81(5)
3.5.1 Exercises for Sect. 3.5
86(1)
3.6 Perron--Wiener Method
86(16)
3.6.1 Exercises for Sect. 3.6
101(1)
3.7 The Radial Limit Theorem
102(4)
3.7.1 Exercises for Sect. 3.7
106(1)
3.8 Nontangential Boundary Limit Theorem
106(8)
3.9 Harmonic Measure
114(7)
References
119(2)
4 Green Functions
121(44)
4.1 Introduction
121(1)
4.2 Green Functions
122(9)
4.2.1 Exercises for Sect. 4.2
131(1)
4.3 Symmetry of the Green Function
131(7)
4.4 Green Potentials
138(13)
4.5 Riesz Decomposition
151(10)
4.5.1 Exercises for Sect. 4.5
161(1)
4.6 Properties of Potentials
161(4)
References
164(1)
5 Negligible Sets
165(50)
5.1 Introduction
165(1)
5.2 Superharmonic Extensions
165(9)
5.2.1 Exercises for Sect. 5.2
174(1)
5.3 Reduction of Superharmonic Functions
174(5)
5.4 Capacity
179(16)
5.4.1 Exercises for Sect. 5.4
195(1)
5.5 Boundary Behavior of the Green Function
195(4)
5.6 Applications
199(7)
5.7 Sweeping
206(9)
References
213(2)
6 Dirichlet Problem for Unbounded Regions
215(44)
6.1 Introduction
215(1)
6.2 Exterior Dirichlet Problem
215(7)
6.3 PWB Method for Unbounded Regions
222(6)
6.4 Boundary Behavior
228(13)
6.5 Intrinsic Topology
241(1)
6.6 Thin Sets
242(4)
6.7 Thinness and Regularity
246(13)
References
258(1)
7 Energy
259(26)
7.1 Introduction
259(1)
7.2 Energy Principle
259(8)
7.3 Mutual Energy
267(8)
7.4 Projections of Measures
275(3)
7.5 Wiener's Test
278(7)
References
284(1)
8 Interpolation and Monotonicity
285(36)
8.1 Introduction
285(1)
8.2 Holder Spaces
286(5)
8.3 Global Interpolation
291(8)
8.4 Interpolation of Weighted Norms
299(3)
8.5 Inner Norms
302(4)
8.6 Monotonicity
306(15)
References
320(1)
9 Newtonian Potential
321(32)
9.1 Introduction
321(1)
9.2 Subnewtonian Kernels
321(14)
9.3 Poisson's Equation
335(2)
9.3.1 Exercises for Sect. 9.3
337(1)
9.4 Holder Continuity of Second Derivatives
337(8)
9.5 The Reflection Principle
345(8)
References
351(2)
10 Elliptic Operators
353(38)
10.1 Introduction
353(1)
10.2 Linear Spaces
354(1)
10.3 Constant Coefficients
355(3)
10.4 Schauder Interior Estimates
358(6)
10.5 Maximum Principles
364(4)
10.6 The Dirichlet Problem for a Ball
368(6)
10.7 Dirichlet Problem for Bounded Domains
374(5)
10.8 Barriers
379(12)
References
390(1)
11 Apriori Bounds
391(22)
11.1 Introduction
391(1)
11.2 Green Function for a Half-Space
391(5)
11.3 Mixed Boundary Conditions for Laplacian
396(10)
11.4 Nonconstant Coefficients
406(7)
References
411(2)
12 Oblique Derivative Problem
413(40)
12.1 Introduction
413(1)
12.2 Boundary Maximum Principle
413(14)
12.3 Curved Boundaries
427(6)
12.4 Superfunctions for Elliptic Operators
433(10)
12.5 Regularity of Boundary Points
443(10)
References
451(2)
13 Application to Diffusion Processes
453(26)
13.1 Introduction
453(1)
13.2 Gaussian Transition Function
453(1)
13.3 Method of Images
454(7)
13.3.1 Exercises for Sect. 13.3
461(1)
13.4 Diffusion as a Random Process
461(6)
13.5 Hille--Yosida Theorem
467(2)
13.6 Diffusions with Reflection and Absorption
469(4)
13.7 Representation as a Family of Kernels
473(6)
References
476(3)
Index 479
The author's interests lie in three interrelated topics: heat equations associated with second-order elliptic operators, Markov or diffusion processes and potential theory.
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