Application of Holomorphic Functions in Two and Higher Dimensions

402 Pages·2016·5.016 MB·English

by  Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig (auth.)

Klaus Gürlebeck
Klaus Habetha
Wolfgang Sprößig
Application
of Holomorphic
Functions in
Two and Higher
Dimensions

Klaus Gürlebeck • Klaus Habetha • Wolfgang Sprößig
Application of Holomorphic
Functions in Two and Higher
Dimensions
Klaus Gürlebeck Klaus Habetha
Bauhaus-U niversität Weimar R WTH Aac hen
Weimar, Germany Aachen, Germany
Wolfgang Sprößig
TU Bergakademie Freiberg
Freiberg, Germany
ISBN 9 78-3-0348-0962-7 ISBN 9 78-3-0348-0964-1 (eBook)
D OI 10.1007/978-3-0348-0964-1
Library of Congress Control Number: 2016942573
Mathematics Subject Classification 2010: 30AXX, 30CXX, 30GXX, 33CXX, 35CXX, 35JXX, 35FXX,
35KXX, 43AXX, 62PXX, 74BXX, 76-XX, 78-XX
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Contents
Preface xi
1 Basic properties of holomorphic functions 1
1.1 Number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Real Clifford numbers . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Quaternion algebra . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 On rotations . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Complex quaternions. . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 Clifford’s geometric algebra . . . . . . . . . . . . . . . . . . 7
1.1.6 The ± split with respect to two square roots of −1 . . . . . 11
1.1.7 Bicomplex numbers . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Classical function spaces in quaternions . . . . . . . . . . . . . . . 16
1.3 New types of holomorphic functions . . . . . . . . . . . . . . . . . 18
1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 Construction of holomorphic functions . . . . . . . . . . . . 21
1.4 Integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 General integral theorems . . . . . . . . . . . . . . . . . . . 23
1.4.2 Integral theorems for holomorphic functions . . . . . . . . . 25
1.5 Polynomial systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.1 Fueter polynomials . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.2 Holomorphic Appell polynomials . . . . . . . . . . . . . . . 33
1.5.3 Holomorphic polynomials for the Riesz system . . . . . . . 34
1.5.4 Orthogonal polynomials in H . . . . . . . . . . . . . . . . . 39
1.5.5 Series expansions . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Conformal and quasi-conformal mappings 43
2.1 Mo¨bius transformations . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.1 Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . 44
2.2 Conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 Conformal mappings in the plane . . . . . . . . . . . . . . 46
2.2.2 Conformal mappings in space . . . . . . . . . . . . . . . . . 47
2.2.3 Mercator projection . . . . . . . . . . . . . . . . . . . . . . 52
v
vi Contents
2.3 Quasi-conformal mappings . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.2 Quaternionic quasi-conformal mappings . . . . . . . . . . . 56
2.4 M-conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.1 Characterization of M-conformal mappings . . . . . . . . . 60
2.4.2 M-conformal mappings in a plane. . . . . . . . . . . . . . . 70
2.4.3 M-conformal mappings of curves on the unit sphere . . . . 72
3 Function theoretic function spaces 75
3.1 Q -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
p
3.2 Properties of Q -spaces . . . . . . . . . . . . . . . . . . . . . . . . 78
p
3.3 Another characterization of Q -spaces . . . . . . . . . . . . . . . . 82
p
3.4 Bergman and Hardy spaces . . . . . . . . . . . . . . . . . . . . . . 89
3.4.1 Bergman space . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.2 Hardy space. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 Riesz potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4 Operator calculus 95
4.1 Teodorescu transform and its left inverse . . . . . . . . . . . . . . . 95
4.1.1 Historical prologue . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.2 Borel–Pompeiu formula . . . . . . . . . . . . . . . . . . . . 96
4.2 On generalized Π-operators . . . . . . . . . . . . . . . . . . . . . . 100
4.2.1 The complex Π-operator . . . . . . . . . . . . . . . . . . . . 100
4.2.2 Shevchenko’s generalization . . . . . . . . . . . . . . . . . . 102
4.2.3 A generalization via the Teodorescu transform . . . . . . . 103
4.2.4 The second generalization of the Π-operator . . . . . . . . . 111
4.2.5 The third generalization of the Π-operator . . . . . . . . . . 114
4.2.6 The special case of quaternions . . . . . . . . . . . . . . . . 117
4.3 A general operator approach to holomorphy . . . . . . . . . . . . . 119
4.3.1 A general holomorphy . . . . . . . . . . . . . . . . . . . . . 120
4.3.2 Types of L-holomorphy . . . . . . . . . . . . . . . . . . . . 122
4.3.3 Taylor type formula . . . . . . . . . . . . . . . . . . . . . . 128
4.3.4 Taylor–Gontcharov formula for generalized Dirac operators
of higher order . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.4 A modified operator calculus in the plane . . . . . . . . . . . . . . 131
4.4.1 Modified Borel–Pompeiu type formulas . . . . . . . . . . . 132
4.4.2 Modified Plemelj–Sokhotski formulas . . . . . . . . . . . . . 134
4.4.3 A modified Dirichlet problem . . . . . . . . . . . . . . . . . 135
4.4.4 A norm estimate for the modified Teodorescu transform . . 137
4.5 Modified operator calculus in space . . . . . . . . . . . . . . . . . . 139
4.5.1 Modified fundamental solutions . . . . . . . . . . . . . . . . 139
4.5.2 A modified Borel–Pompeiu formula. . . . . . . . . . . . . . 142
4.6 Operator calculus on the sphere . . . . . . . . . . . . . . . . . . . . 144
4.6.1 Gegenbauer functions . . . . . . . . . . . . . . . . . . . . . 144
Contents vii
4.6.2 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . 145
4.6.3 Borel–Pompeiu formula . . . . . . . . . . . . . . . . . . . . 149
5 Decompositions 151
5.1 Vector fields in Euclidean space . . . . . . . . . . . . . . . . . . . . 151
5.1.1 Helmholtz decomposition . . . . . . . . . . . . . . . . . . . 151
5.1.2 Associated boundary value problems . . . . . . . . . . . . . 154
5.1.3 Original Hodge decomposition theorem . . . . . . . . . . . 155
5.2 Bergman–Hodge decompositions . . . . . . . . . . . . . . . . . . . 156
5.2.1 Suitable fundamental solutions . . . . . . . . . . . . . . . . 157
5.2.2 An orthogonal decomposition formula with complex potential159
5.2.3 Generalized Bergman–Hodge decomposition . . . . . . . . . 162
5.2.4 Decompositions in domains on the unit sphere . . . . . . . 162
5.3 Representations of functions by holomorphic generators . . . . . . 164
5.3.1 Almansi decomposition . . . . . . . . . . . . . . . . . . . . 164
5.3.2 Fischer decomposition . . . . . . . . . . . . . . . . . . . . . 166
6 Some first-order systems of partial differential equations 169
6.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.1 A brief historical review . . . . . . . . . . . . . . . . . . . . 169
6.1.2 Stationary Maxwell equations . . . . . . . . . . . . . . . . . 172
6.1.3 Stationary Maxwell equations with variable permitivities . 173
6.2 Bers-Vekua systems . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2.1 History of the Vekua equation. . . . . . . . . . . . . . . . . 175
6.2.2 Pseudoanalytic functions . . . . . . . . . . . . . . . . . . . 177
6.2.3 Generating sequences and formal powers . . . . . . . . . . . 179
6.2.4 An important special case . . . . . . . . . . . . . . . . . . . 181
6.2.5 Orthogonal coordinates and explicit generating sequences . 182
6.2.6 Completeness of the systems of formal powers . . . . . . . . 184
6.2.7 Factorization of second-order operators in the plane . . . . 185
6.2.8 CompletesystemsofsolutionsforthestationarySchro¨dinger
equation and their applications . . . . . . . . . . . . . . . . 188
6.2.9 The Riccati equation in two dimensions . . . . . . . . . . . 190
6.2.10 On the solution of the Riccati equation . . . . . . . . . . . 191
6.2.11 Factorization in the hyperbolic case . . . . . . . . . . . . . 194
6.3 Biquaternions and factorization of spatial models . . . . . . . . . . 195
6.3.1 Biquaternionic Vekua-type equations from physics . . . . . 195
6.3.2 Factorization of the 3D-Schro¨dinger operator and the main
biquaternionic Vekua equation . . . . . . . . . . . . . . . . 197
7 Boundary value problems for second-order partial differential equations 203
7.1 p-harmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.1.1 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . 203
7.1.2 p-harmonic functions . . . . . . . . . . . . . . . . . . . . . . 207
viii Contents
7.2 A class of non-linear boundary value problems . . . . . . . . . . . 208
7.3 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.3.1 Motivation and historical note . . . . . . . . . . . . . . . . 210
7.3.2 Square roots of the Helmholtz operator . . . . . . . . . . . 212
7.4 Yukawa’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.4.1 An operator theory . . . . . . . . . . . . . . . . . . . . . . . 219
7.5 Equations of linear elasticity. . . . . . . . . . . . . . . . . . . . . . 222
7.5.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.5.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.5.3 Solution theory for the stationary problem. . . . . . . . . . 230
7.5.4 Kolosov-Muskhelishvili formulas . . . . . . . . . . . . . . . 232
7.5.5 Fundamentals of the linear theory of elasticity . . . . . . . 234
7.5.6 General solution of Papkovic-Neuber . . . . . . . . . . . . . 236
7.5.7 The representation theorem of Goursat in H. . . . . . . . . 237
7.5.8 Spatial Kolosov-Muskhelishvili formulas in H . . . . . . . . 239
7.5.9 Generalized Kolosov-Muskhelishvili formulas for stresses . . 241
7.6 Transmission problems in linear elasticity . . . . . . . . . . . . . . 247
7.6.1 Boundary value problems in multiply connected domains . 248
7.6.2 Solution of the transmission problem . . . . . . . . . . . . . 250
7.6.3 Transmission problems for the Lam´e system . . . . . . . . . 254
7.7 Stationary fluid flow problems . . . . . . . . . . . . . . . . . . . . . 254
7.7.1 A brief history of fluid dynamics . . . . . . . . . . . . . . . 254
7.7.2 Stationary linear Stokes problem . . . . . . . . . . . . . . . 255
7.7.3 Non-linear Stokes equations . . . . . . . . . . . . . . . . . . 256
7.7.4 Stationary Navier-Stokes problem . . . . . . . . . . . . . . 257
7.7.5 Stationary equations of thermo-fluid dynamics . . . . . . . 258
7.7.6 Stationary magneto-hydromechanics . . . . . . . . . . . . . 259
8 Some initial-boundary value problems 265
8.1 Rothe’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.2 Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.3 Galpern-Sobolev equations. . . . . . . . . . . . . . . . . . . . . . . 269
8.3.1 Description of the problem . . . . . . . . . . . . . . . . . . 270
8.3.2 Quaternionic integral operators . . . . . . . . . . . . . . . . 273
8.3.3 A representation formula . . . . . . . . . . . . . . . . . . . 274
8.4 The Poisson-Stokes problem . . . . . . . . . . . . . . . . . . . . . . 276
8.4.1 Semi–discretization . . . . . . . . . . . . . . . . . . . . . . . 278
8.4.2 Operator decomposition . . . . . . . . . . . . . . . . . . . . 280
8.4.3 Representation formulas . . . . . . . . . . . . . . . . . . . . 280
8.5 Higher dimensional versions of Korteweg-de Vries’ and Burgers’
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.5.1 Multidimensional version of Burgers equation . . . . . . . . 282
8.5.2 Airy’s equation . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.5.3 A quaternionic Korteweg-de Vries-Burgers equation . . . . 287
Contents ix
8.6 Solving the Maxwell equations . . . . . . . . . . . . . . . . . . . . 288
8.7 Alternative treatment of parabolic problems . . . . . . . . . . . . . 290
8.7.1 The Witt basis approach . . . . . . . . . . . . . . . . . . . 290
8.7.2 Harmonic extension method . . . . . . . . . . . . . . . . . . 292
8.8 Fluid flow through porous media . . . . . . . . . . . . . . . . . . . 294
8.8.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . 294
8.8.2 Representation in a quaternionic operator calculus . . . . . 295
8.8.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9 Riemann-Hilbert problems 303
9.1 Riemann-Hilbert problem in the plane . . . . . . . . . . . . . . . . 303
9.2 Riemann-Hilbert problems in C(cid:2)(3,0) . . . . . . . . . . . . . . . . 307
9.2.1 Plemelj formula for functions with a parameter . . . . . . . 308
9.2.2 Riemann boundary value problem for harmonic functions . 316
9.2.3 Riemann boundary value problem for biharmonic functions 317
10 Initial-boundary value problems on the sphere 319
10.1 Forecasting equations . . . . . . . . . . . . . . . . . . . . . . . . . 319
10.1.1 Forecasting equations – a physical description . . . . . . . . 319
10.1.2 Toroidal flows on the sphere . . . . . . . . . . . . . . . . . . 321
10.1.3 Tangential derivatives . . . . . . . . . . . . . . . . . . . . . 322
10.1.4 Oseen’s problem on the sphere . . . . . . . . . . . . . . . . 323
10.1.5 Forecasting equations in a ball shell . . . . . . . . . . . . . 325
10.2 Viscous shallow water equations. . . . . . . . . . . . . . . . . . . . 326
11 Fourier transforms 329
11.1 Hypercomplex Fourier transforms . . . . . . . . . . . . . . . . . . . 329
11.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11.1.2 General two-sided Clifford Fourier transforms . . . . . . . . 330
11.1.3 Properties of the general two-sided CFT . . . . . . . . . . . 331
11.1.4 Fourier transforms in quaternions . . . . . . . . . . . . . . . 334
11.1.5 Clifford Fourier-Mellin transform . . . . . . . . . . . . . . . 340
11.1.6 Clifford–Fourier transforms with pseudoscalar square roots
of −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.1.7 Spacetime Fourier transform . . . . . . . . . . . . . . . . . 343
11.1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
11.2 Fractional Fourier transform . . . . . . . . . . . . . . . . . . . . . . 346
11.2.1 Exponentials of the Dirac operator . . . . . . . . . . . . . . 346
11.2.2 Fourier transform of fractional order . . . . . . . . . . . . . 348
11.3 Radon transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
11.3.1 A basic problem . . . . . . . . . . . . . . . . . . . . . . . . 351
11.3.2 At the very beginning . . . . . . . . . . . . . . . . . . . . . 351
11.3.3 Passing to higher dimensions . . . . . . . . . . . . . . . . . 352
11.3.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 352