Singular Nonlinear Partial Differential Equations

Das Anliegen dieser mathematischen Monographie ist die Zusammenfassung aller Resultate, die heutzutage vorliegen über die Existenz formaler, holomorpher oder singulärer Lösungen von singulären nicht-linearen partiellen Differentialgleichungen.

Autorentext

Prof. Raymond Gerard ist am Institut de Recherche Mathématique Alsacien an der Université Louis Pasteur in Strasbourg beschäftigt. Prof. Hidetoshi Tahara lehrt an der Sophia Universität in Tokyo.

Inhalt
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 ?[].- 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 's type theorems for non linear singular partial differential equations.- 6.1 Implicit function theorem.- 6.2 Nonlinear 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 Examples.- 6.5 Supplementary results.- 7 Maillet's 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 Maillet's type theorems for non linear singular integrodifferential 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.