Boundary Integral Equations on Contours with Peaks

Contents......Page 6
Preface......Page 9
1.1 Introduction......Page 12
1.2.1 Auxiliary assertions......Page 16
1.2.2 Estimates for kernels of integral operators......Page 20
1.2.3 Single and double layer potentials on contours with peak......Page 27
1.2.4 Operator πI......Page 34
1.2.5 Operator πI......Page 40
1.3.1 Dirichlet problem for a domain with outward peak......Page 45
1.3.2 Boundary value problems for a domain with inward peak......Page 51
1.3.3 Auxiliary boundary value problems for a domain with peak......Page 54
1.4.1 Integral equations of the first kind......Page 71
1.4.2 Integral equation of the interior Dirichlet problem in a domain with outward peak......Page 79
1.4.3 Integral equation of the interior Neumann problem in a domain with outward peak......Page 84
1.4.4 Integral equations of the exterior Dirichlet and Neumann problems in a domain with inward peak......Page 89
1.4.5 Boundary integral equation of the Dirichlet problem in a domain with inward peak......Page 91
1.4.6 Boundary integral equation of the Neumann problem in a domain with inward peak......Page 96
1.4.7 Integral equations of the exterior Dirichlet and Neumann problems for domain with outward peak......Page 103
1.5 Direct method of integral equations of the Neumann and Dirichlet problems......Page 106
2 Boundary Integral Equations inH¨older Spaces on a Contour with Peak......Page 112
2.1 Weighted H¨older spaces......Page 114
2.2.1 Integral operators in weighted H¨older spaces......Page 117
2.2.2 Continuity of the operator πI......Page 121
2.2.3 Continuity of the operator πI + S......Page 131
2.2.4 Continuity of the operator......Page 138
2.3 Dirichlet and Neumann problems in a strip......Page 145
2.4.1 Auxiliary assertions......Page 162
2.4.2 Integral equation of the Dirichlet problem on a contour with outward peak......Page 168
2.4.3 Integral equation of the Neumann problem on a contour with outward peak......Page 177
2.5.1 Integral equation of the Dirichlet problem on a contour with inward peak......Page 186
2.5.2 Integral equation of the Neumann problem on contour with inward peak......Page 190
2.6 Integral equation of the first kind on a contour with peak......Page 198
Appendix A: Proof of Theorem 2.2.1......Page 204
Appendix B: Proof of Corollary 2.2.2......Page 209
Appendix C: To proof of Theorem 2.2.7......Page 211
Appendix D: To proof of Theorem 2.2.9......Page 216
Appendix E: To proof of Theorem 2.2.12......Page 218
3 Asymptotic Formulae forSolutions of Boundary IntegralEquations Near Peaks......Page 226
3.1.1 Asymptotics of a conformal mapping of a domain with outward peak onto a strip......Page 230
3.1.2 Asymptotics of a conformal mapping of a domain with inward peak onto the upper half-plane......Page 234
3.2 The Dirichlet and Neumann problems in domains with peaks......Page 237
3.3.1 Homogeneous integral equation of the problem......Page 246
3.3.2 Solvability of the integral equation of the problem......Page 255
3.3.3 Integral equation of the problem......Page 258
3.4.1 Homogeneous integral equation of the problem......Page 259
3.4.2 Solvability of the integral equation of the problem......Page 261
3.4.3 Integral equation of the problem......Page 264
Appendix A: Counterexample......Page 265
Appendix B: Proof of Lemma 3.2.1......Page 267
Appendix C: Proof of Lemma 3.2.2......Page 270
Appendix D: Proof of Lemma 3.2.3......Page 276
Appendix E: Proof of Lemma 3.2.4......Page 281
4.1 Introduction......Page 284
4.2.1 Asymptotic behavior of solutions to the problem......Page 292
4.2.2 Asymptotic behavior of solutions to the problem......Page 304
4.2.3 Properties of solutions to the problem......Page 321
4.2.4 Uniqueness theorems......Page 323
4.3.1 Integral equations of the problem......Page 324
4.3.2 Integral equation of the problem......Page 334
4.4.1 Integral equation of the problem......Page 337
4.4.2 Integral equation of the problem......Page 342
Bibliography......Page 346
List of Symbols......Page 350
Index......Page 352