Author | Gรฉrard, Raymond. author |
---|---|
Title | Singular Nonlinear Partial Differential Equations [electronic resource] / by Raymond Gรฉrard, Hidetoshi Tahara |
Imprint | Wiesbaden : Vieweg+Teubner Verlag, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-322-80284-2 |
Descript | VIII, 272 p. online resource |
1 Operators with regular singularities: One variable case -- 1.1 Notations, definitions, examples -- 1.2 The good operators -- 1.3 A class of operators with a regular singularity -- 1.4 Applications to differential equations -- 1.5 The Maillet theorem -- 2 Operators with regular singularities: Several variables case -- A Formal theory -- 2.1 Notations -- 2.2 Linear operators on ?[[x]] -- 2.3 Non linear operators on ? f -- 2.4 Solutions of linear equations -- 2.5 Solutions of non linear equations -- B Analytic theory -- 2.6 Notations and definitions -- 2.7 The good operators and the notion of domination -- 2.8 A class of operators having a regular singularity -- 2.9 Applications to partial differential equations -- 3 Formal and convergent solutions of singular partial differential equations -- 3.1 Notations and definitions -- 3.2 Holomorphic solutions of certain equations -- 3.3 Equations with parameters -- 3.4 An application: A theorem of S. Kaplan -- 3.5 The case of small denominators -- 4 Local study of differential equations of the form xy? = f(x,y) near x = 0 -- 4.1 Coupling of two differential equations -- 4.2 Behavior of solutions of a differential equation near a regular point -- 4.3 Local study of a differential equation near a singular point of regular type -- 4.4 Study of the Hukuhara equation and of the Hukuhara function -- 5 Holomorphic and singular solutions of non linear singular first order partial differential equations -- 5.1 Notations and definitions -- 5.2 Statement of the main theorem -- 5.3 Holomorphic solutions -- 5.4 Singular solutions -- 5.5 Uniqueness of the solution -- 5.6 Proof of the main theorem 5.2.3 -- 5.7 Remarks -- 5.8 Supplementary result -- 6 Maillet โs type theorems for non linear singular partial differential equations -- 6.1 Implicit function theorem -- 6.2 Non linear equations with first order linear part -- 6.3 Non linear equations with higher order linear part -- 6.4 Formal Gevrey index for a particular type of equations โ Examples -- 6.5 Supplementary results -- 7 Mailletโs type theorems for non linear singular partial differential equations without linear part -- 7.1 Notations and definitions -- 7.2 Assumptions and results -- 7.3 A basic lemma -- 7.4 Proof of theorem 7.2.5 -- 7.5 Complementary results -- 7.6 A remark -- 8 Holomorphic and singular solutions of non linear singular partial differential equations -- 8.1 Holomorphic solutions -- 8.2 Singular solutions: Special case -- 8.3 Singular solutions: General case -- 8.4 Asymptotic study -- 8.5 Completion of the proof of the main theorem -- 9 On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations -- 9.1 Notations and definitions -- 9.2 Results -- 9.3 Proof of theorem 9.2.1 -- 9.4 Proof of theorem 9.2.3 -- 10 Mailletโs type theorems for non linear singular integroโdifferential equations -- 10.1 Notations and definitions -- 10.2 The main theorem -- 10.3 Construction of the formal solution -- 10.4 Some discussions -- 10.5 Convergence of the formal solution in the case sl = 1 -- 10.6 Convergence of the formal solution in the case sl > 1 -- 10.7 Supplementary results and remark