Elliptic, hyperbolic and mixed complex equations with parabolic degeneracy

✍ Scribed by Guo Chun Wen

• Publisher: World Scientific Publishing Company
• Year: 2007
• Tongue: English
• Leaves: 453
• Series: Peking University Series in Mathematics
• Category: Library

✦ Synopsis

In the recent half-century, many mathematicians have investigated various problems on several equations of mixed type and obtained interesting results, with important applications to gas dynamics. However, the Tricomi problem of general mixed type equations of second order with parabolic degeneracy has not been completely solved, particularly the Tricomi and Frankl problems for general Chaplygin equation in multiply connected domains posed by L Bers, and the existence, regularity of solutions of the above problems for mixed equations with non-smooth degenerate curve in several domains posed by J M Rassias.

The method revealed in this book is unlike any other, in which the hyperbolic number and hyperbolic complex function in hyperbolic domains, and the complex number and complex function in elliptic domains are used. The corresponding problems for first order complex equations with singular coefficients are first discussed, and then the problems for second order complex equations are considered, where we pose the new partial derivative notations and complex analytic methods such that the forms of the above first order complex equations in hyperbolic and elliptic domains are wholly identical. In the meantime, the estimates of solutions for the above problems are obtained, hence many open problems including the above Tricomi Bers and Tricomi Frankl Rassias problems can be solved.

Contents: Elliptic Complex Equations of First Order; Elliptic Complex Equations of Second Order; Hyperbolic Complex Equations of First and Second Orders; First Order Complex Equations of Mixed Type; Second Order Linear Equations of Mixed Type; Second Order Quasilinear Equations of Mixed Type.

✦ Table of Contents

Contents......Page 12
Preface......Page 6
1.1 Reduction of general uniformly elliptic systems of rst order equations to standard complex form......Page 15
1.2 Representation of solutions of discontinuous Riemann-Hilbert problem for elliptic complex equations......Page 19
1.3 Existence of solutions of discontinuous Riemann-Hilbert problem for nonlinear complex equations in upper half-unit disk......Page 22
1.4 The discontinuous Riemann-Hilbert problem for nonlinear complex equations in general domains......Page 23
2.1 Formulation of the Riemann-Hilbert problem for degenerate elliptic complex equations......Page 25
2.2 Representations and estimates of solutions of Riemann-Hilbert problem for elliptic complex equations......Page 28
2.3 Solvability of Riemann-Hilbert problem for degenerate elliptic complex equations......Page 35
3.1 Formulation of discontinuous Riemann- Hilbert problem for degenerate elliptic complex equations......Page 38
3.2 Representation and uniqueness of solutions of discontinuous Riemann-Hilbert problem for elliptic complex equations......Page 40
3.3 Estimates and existence of solutions of Riemann-Hilbert problem for degenerate elliptic complex equations......Page 42
4.1 Formulation of Riemann-Hilbert problem for degenerate elliptic complex equations in multiply connected domains......Page 45
4.2 Representation and uniqueness of solutions of Riemann-Hilbert problem for degenerate elliptic complex equations......Page 47
4.3 Estimates of solutions of Riemann-Hilbert problem for degenerate elliptic equations......Page 49
4.4 Existence of solutions of Riemann-Hilbert problem for degenerate elliptic equations......Page 51
1.1 Formulation of discontinuous oblique derivative problem for elliptic equations......Page 55
1.2 The representation theorem of discontinuous oblique derivative problem for elliptic equations......Page 59
1.3 Existence of solutions of discontinuous oblique derivative problem for elliptic equations in upper half-unit disk......Page 66
1.4 The discontinuous oblique derivative problem for elliptic equations in general domains......Page 70
2.1 Formulation of mixed boundary value problem for degenerate elliptic equations of second order......Page 73
2.2 Representation of solutions of mixed boundary value problem for degenerate elliptic equations......Page 80
2.3 Estimates and existence of solutions of mixed problem for degenerate elliptic equations......Page 82
3.1 Formulation of oblique derivative problem for degenerate elliptic equations......Page 94
3.2 Representation of solutions of oblique derivative problem for elliptic equations......Page 98
3.3 Estimates of solutions of oblique derivative problem for elliptic equations......Page 102
3.4 Existence of solutions of oblique derivative problem for elliptic equations......Page 111
4.1 Boundary value problems for homogeneous elliptic equations of second order with degenerate rank 0......Page 115
4.2 Boundary value problems for axisymmetric ltration......Page 123
5. The Oblique Derivative Problem for Nonhomogeneous Elliptic Equations of Second Order with Degenerate Rank 0......Page 127
5.1 Formulation of oblique derivative problems for degenerate elliptic equations......Page 128
5.2 Representation of solutions of oblique derivative problem of elliptic equations......Page 130
5.3 Estimates and existence of solutions of oblique derivative derivative problems......Page 132
1.1 Complex forms of linear and quasilinear hyperbolic system of rst order equations......Page 143
1.2 Formulation of the Riemann-Hilbert problem and uniqueness of its solutions for simplest hyperbolic complex equation......Page 146
1.3 Uniqueness of solutions of the Riemann-Hilbert problem for linear hyperbolic complex equations......Page 148
1.4 Solvability of Riemann-Hilbert problem for linear hyperbolic complex equations......Page 151
1.5 Another boundary value problem for linear hyperbolic complex equations of rst order......Page 153
2.1 Formulation of Riemann-Hilbert problem for degenerate hyperbolic complex equations of rst order......Page 155
2.2 Representation of solutions of Riemann- Hilbert problem for degenerate hyperbolic complex equations......Page 157
2.3 Existence and uniqueness of solutions of Riemann-Hilbert problem......Page 158
2.4 Unique solvability of Cauchy problem for degenerate hyperbolic complex equations......Page 160
3.1 Complex forms of hyperbolic equations of second order......Page 165
3.2 Formulation of oblique derivative problem and representations of solutions......Page 168
3.3 Existence and uniqueness of solutions of oblique derivative problems......Page 174
4. The Oblique Derivative Problem for Degenerate Hyperbolic Equations of Second Order......Page 176
4.1 Formulation of discontinuous oblique derivative problem for degenerate hyperbolic equations......Page 177
4.2 Representation and solvability of oblique derivative problem for degenerate hyperbolic equations......Page 179
4.3 Oblique derivative problem for degenerate hyperbolic equations in general domains......Page 187
5.1 Formulation of oblique derivative problem for second order hyperbolic equations......Page 192
5.2 Representations of solutions of oblique derivative problems for second order hyperbolic equations......Page 193
5.3 Unique solvability of oblique derivative problem for second order hyperbolic equations......Page 200
5.4 Oblique derivative problems for quasilinear hyperbolic equations with degenerate rank 0......Page 202
6. The Cauchy Problem for Hyperbolic Equations of Second Order with Degenerate Rank 0......Page 203
6.1 Formulation of Cauchy problem for second order hyperbolic equations......Page 204
6.2 Reduction of Cauchy problem for degenerate hyperbolic equations to integral equations......Page 205
6.3 Existence of solutions of Cauchy problem for degenerate hyperbolic equations......Page 208
1.1 Formulation of Riemann-Hilbert problem of complex equations of mixed type......Page 217
1.2 Representation of Riemann-Hilbert problem for mixed complex equations......Page 219
1.3 Unique solvability of Riemann-Hilbert problem for complex equations of mixed type......Page 222
2.1 Formulation of Riemann-Hilbert problem for linear degenerate mixed complex equations......Page 228
2.2 Representation and uniqueness of solutions of Riemann-Hilbert problem......Page 230
2.3 Solvability of Riemann-Hilbert problem for degenerate mixed equations......Page 236
3.1 Representation and uniqueness of solutions of discontinuous Riemann-Hilbert problem......Page 243
3.2 Riemann-Hilbert problem for quasilinear mixed equations in general domains......Page 246
4.1 Formulation of general boundary value problem for complex equations of mixed type......Page 251
4.2 Representation of solutions for general boundary value problem......Page 254
4.3 Solvability of general boundary value problem for degenerate mixed equations......Page 258
1.1 Formulation of oblique derivative problem for second order equations of mixed type......Page 261
1.2 Representation and uniqueness of solutions for oblique derivative problem......Page 264
1.3 The solvability of oblique derivative problem for second order equations of mixed type......Page 269
2. The Tricomi Problem for Second Order Degenerate Equations of Mixed Type......Page 272
2.1 The Tricomi problem of second order degenerate equations of mixed type......Page 273
2.2 Representation and uniqueness of solutions of Tricomi problem for degenerate mixed equations......Page 277
2.3 Existence of solutions of Tricomi problem for degenerate equations of mixed type......Page 285
3.1 Formulation of discontinuous oblique derivative problem for equations of mixed type......Page 296
3.2 Representation of solutions of discontinuous oblique derivative problem for mixed equations......Page 300
3.3 Solvability of discontinuous oblique derivative problem for degenerate equations of mixed type......Page 305
4. The Exterior Tricomi-Rassias Problem for Second Order Degenerate Equations of Mixed Type......Page 310
4.1 Formulation of exterior Tricomi-Rassias problem for degenerate equations of mixed type......Page 311
4.2 Representation of solutions of exterior Tricomi-Rassias Problem......Page 317
4.3 Unique solvability of solutions of exterior Tricomi-Rassias problem......Page 324
5.1 Formulation of Frankl problem for second order equations of mixed type......Page 332
5.2 Representation and uniqueness of solutions of Frankl problem for degenerate mixed equations......Page 337
5.3 Existence of solutions of Frankl problem for degenerate equations of mixed type......Page 341
1.1 Formulation of oblique derivative problem for second order equations of mixed type......Page 347
1.2 Existence and uniqueness of solutions for oblique derivative problem......Page 349
1.3 C1 ( D )-estimate of solutions of oblique derivative problem for second order mixed equations......Page 351
2.1 Formulation of oblique derivative problem for mixed equations with parabolic degeneracy......Page 354
2.2 Representation of solutions of oblique derivative problem for degenerate mixed equations......Page 359
2.3 Existence of solutions of oblique derivative problem for degenerate mixed equations......Page 365
2.4 Oblique derivative problem for degenerate mixed equations in general domains......Page 370
3.1 Formulation of oblique derivative problem in multiply connected domains......Page 375
3.2 Representation of solutions of Tricomi problem for degenerate equations of mixed type......Page 381
3.3 Uniqueness of solutions of Tricomi problem for degenerate equation of mixed type......Page 385
3.4 Solvability of Tricomi problem for degenerate equation of mixed type......Page 387
4.1 Formulation of oblique derivative problem for mixed equations with nonsmooth degeneracy......Page 398
4.2 Representation of solutions of oblique derivative problem for mixed equations......Page 403
4.3 Existence of solutions of oblique derivative problem for mixed equations......Page 409
5.1 Formulation of oblique derivative problem for mixed equations with degenerate rank 0......Page 412
5.2 Representation of solutions of oblique derivative problem for degenerate mixed equations......Page 416
5.3 Existence of solutions of oblique derivative problem for degenerate mixed equations......Page 421
References......Page 429
Index......Page 443