Shargorodsky E. Bernoulli free-boundary problems / E.Shargorodsky, J.F.Toland. - Providence: American Mathematical Society, 2008. - vii, 70 p.: ill. - (Memoirs of the American Mathematical Society; Vol.196, N 914). - Bibliogr.: p.65-67. - Ind.: p.69-70. - ISBN 978-0-8218-4189-1; ISSN 0065-9266  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Chapter 1. Introduction ........................................ 1 Chapter 2. Bernoulli Free Boundaries ........................... 3 2.1. Special case: steady hydrodynamic waves ................ 4 2.2. General Case ........................................... 4 2.3. Notation ............................................... 7 2.4. Formulation as a Single Equation ....................... 9 2.5. Equations .............................................. 9 2.6. Example of (2.7) with Explicit Solutions .............. 10 2.7. Equivalence ........................................... 11 2.8. Inequalities .......................................... 12 2.9. Duality ............................................... 13 2.10. Example of (2.7) with Explicit Solutions: Duality ..... 13 2.11. Self-duality .......................................... 14 Chapter 3. Type-(I) Problems .................................. 15 3.1. Regularity ............................................ 15 3.2. Example of (2.7) with Explicit Solutions: Regularity ............................................ 16 3.3. Dimension of the Set of Stagnation Points ............. 16 3.4. Jordan Curves ......................................... 16 3.5. Example of (2.7) with Explicit Solutions: Jordan Curves ................................................ 17 3.6. Nekrasov's Equation ................................... 17 3.7. Nekrasov Duality ...................................... 19 3.8. Example of (2.7) with Explicit Solutions: Nekrasov Duality ............................................... 20 3.9. Morse Index of Non-singular Solutions ................. 20 3.10. Example of (2.7) with Explicit Solutions: Morse Index ................................................. 23 3.11. Stokes Waves .......................................... 23 Chapter 4. Proofs of Main Results ............................. 27 4.1. Equations: proofs of Theorem 2.4 and Corollary 2.5 .... 27 4.2. Equivalence: proofs of Theorems 2.7, 2.8 and 2.9 ...... 28 4.3. Inequalities: proof of Theorem 2.10 ................... 34 4.4. Duality ............................................... 35 4.5. Regularity: proofs of Theorems 3.1 and 3.3 ............ 38 4.6. Dimension of the Set of Stagnation Points: proof of Theorem ............................................... 41 4.7. Jordan Curves: proofs of Theorem 3.5 and (3.3) ........ 42 4.8. Nekrasov's Equation: proof of Theorem 3.8 ............. 48 4.9. Morse Indices ......................................... 49 4.10. Plotnikov's Transformation ............................ 50 4.11. Sign of the Plotnikov Potential ....................... 53 4.12. Constant Plotnikov Potentials: Proofs of Theorem 3.14 .................................................. 53 4.13. Simple Morse-Index Estimates: Proof of Lemma 3.10 ..... 54 4.14. Morse Index and Stagnation Points ..................... 54 4.15. Proof of Theorem 3.13 ................................. 61 Appendix A. Auxiliary results ................................. 63 Bibliography ................................................... 65 Index .......................................................... 69