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Introduction to Heat Potential Theory [Kõva köide]

  • Formaat: Hardback, 266 pages, kaal: 643 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-May-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821849980
  • ISBN-13: 9780821849989
Teised raamatud teemal:
Introduction to Heat Potential TheorySuurem pilt
  • Formaat: Hardback, 266 pages, kaal: 643 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-May-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821849980
  • ISBN-13: 9780821849989
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780821849989.html
Märksõnad:
This book is the first to be devoted entirely to the potential theory of the heat equation, and thus deals with time dependent potential theory. Its purpose is to give a logical, mathematically precise introduction to a subject where previously many proofs were not written in detail, due to their similarity with those of the potential theory of Laplace's equation. The approach to subtemperatures is a recent one, based on the Poisson integral representation of temperatures on a circular cylinder. Characterizations of subtemperatures in terms of heat balls and modified heat balls are proved, and thermal capacity is studied in detail. The generalized Dirichlet problem on arbitrary open sets is given a treatment that reflects its distinctive nature for an equation of parabolic type. Also included is some new material on caloric measure for arbitrary open sets. Each chapter concludes with bibliographical notes and open questions. The reader should have a good background in the calculus of functions of several variables, in the limiting processes and inequalities of analysis, in measure theory, and in general topology for Chapter 9.
Preface vii
Notation and Terminology xi
Chapter 1 The Heat Operator, Temperatures and Mean Values
1(34)
1.1 Temperatures and Heat Balls
1(2)
1.2 Mean Values of Smooth Functions over Heat Spheres
3(4)
1.3 Mean Values of Smooth Subtemperatures over Heat Spheres
7(6)
1.4 Mean Values of Smooth Subtemperatures over Heat Balls
13(4)
1.5 The Boundary Maximum Principle on Circular Cylinders
17(2)
1.6 Modified Heat Balls
19(6)
1.7 Harnack Theorems
25(4)
1.8 Equicontinuous Families of Temperatures
29(2)
1.9 Notes and Comments
31(4)
Chapter 2 The Poisson Integral for a Circular Cylinder
35(18)
2.1 The Cauchy Problem on a Half-Space
35(2)
2.2 The Dirichlet Problem on a Circular Cylinder
37(2)
2.3 Double Layer Heat Potentials
39(5)
2.4 The Poisson Integral and the Caloric Measure
44(3)
2.5 Characterizations of Temperatures
47(4)
2.6 Extensions of some Harnack Theorems
51(1)
2.7 Notes and Comments
52(1)
Chapter 3 Subtemperatures and the Dirichlet Problem on Convex Domains of Revolution
53(32)
3.1 Semicontinuous Functions
53(2)
3.2 Subtemperatures
55(9)
3.3 The Dirichlet Problem on Convex Domains of Revolution
64(5)
3.4 Boundary Behaviour of the PWB Solution
69(2)
3.5 Characterizations of Hypotemperatures and Subtemperatures
71(9)
3.6 Properties of Hypotemperatures
80(2)
3.7 Thermic Majorants
82(1)
3.8 Notes and Comments
83(2)
Chapter 4 Temperatures on an Infinite Strip
85(20)
4.1 An Extension of the Maximum Principle on an Infinite Strip
85(2)
4.2 Gauss-Weierstrass Integrals
87(8)
4.3 Nonnegative Temperatures
95(6)
4.4 Minimality of the Fundamental Temperature
101(2)
4.5 Notes and Comments
103(2)
Chapter 5 Classes of Subtemperatures on an Infinite Strip
105(22)
5.1 Hyperplane Mean Values and Classes of Subtemperatures
105(9)
5.2 Behaviour of the Hyperplane Mean Values of Subtemperatures
114(5)
5.3 Classes of Subtemperatures and Nonnegative Thermic Majorants
119(4)
5.4 Characterizations of the Gauss-Weierstrass Integrals of Functions
123(3)
5.5 Notes and Comments
126(1)
Chapter 6 Green Functions and Heat Potentials
127(32)
6.1 Green Functions
127(4)
6.2 Green Functions and the Adjoint Heat Equation
131(3)
6.3 Heat Potentials
134(6)
6.4 The Distributional Heat Operator
140(6)
6.5 The Riesz Decomposition Theorem
146(4)
6.6 Monotone Approximation by Smooth Supertemperatures
150(1)
6.7 Further Characterizations of Subtemperatures
151(1)
6.8 Supertemperatures on an Infinite Strip or Half-Space
152(5)
6.9 Notes and Comments
157(2)
Chapter 7 Polar Sets and Thermal Capacity
159(36)
7.1 Polar Sets
159(3)
7.2 Families of Supertemperatures
162(4)
7.3 The Natural Order Decomposition
166(4)
7.4 Reductions and Smoothed Reductions
170(5)
7.5 The Thermal Capacity of Compact Sets
175(3)
7.6 The Thermal Capacity of More General Sets
178(5)
7.7 Thermal and Cothermal Capacities
183(1)
7.8 Capacitable Sets
183(4)
7.9 Polar Sets and Heat Potentials
187(1)
7.10 Thermal Capacity and Lebesgue Measure
188(4)
7.11 Notes and Comments
192(3)
Chapter 8 The Dirichlet Problem on Arbitrary Open Sets
195(36)
8.1 Classification of Boundary Points
196(3)
8.2 Upper and Lower PWB Solutions
199(6)
8.3 Resolutivity and PWB Solutions
205(2)
8.4 The Caloric Measure on the Essential Boundary
207(7)
8.5 Boundary Behaviour of PWB Solutions
214(8)
8.6 Geometric Tests for Regularity
222(3)
8.7 Green Functions, Heat Potentials, and Thermal Capacity
225(3)
8.8 Notes and Comments
228(3)
Chapter 9 The Thermal Fine Topology
231(28)
9.1 Definitions and Basic Properties
231(6)
9.2 Further Properties of Reductions
237(3)
9.3 The Fundamental Convergence Theorem
240(4)
9.4 Applications of the Fundamental Convergence Theorem to Reductions
244(5)
9.5 Thermal Thinness and the Regularity of Normal Boundary Points
249(3)
9.6 Thermal Fine Limits and Euclidean Limits
252(1)
9.7 Thermal Thinness and the Quasi-Lindelof Property
253(4)
9.8 Notes and Comments
257(2)
Bibliography 259(4)
Index 263