Operators with regular singularities. One variable case - Operators with regular singularities. Several variables case - Formal and convergent solutions of singular partial differential equations - Local study of differential equations of the form xy' f(x,y) near x 0 - Holomorphic and singular solutions of non linear singular first order partial differential equations - Maillet's type theorems for non linear singular partial differential equations - Maillet's type theorems for non linear singular partial differential equations without linear part - Holomorphic and singular solutions of non…mehr
Operators with regular singularities. One variable case - Operators with regular singularities. Several variables case - Formal and convergent solutions of singular partial differential equations - Local study of differential equations of the form xy' f(x,y) near x 0 - Holomorphic and singular solutions of non linear singular first order partial differential equations - Maillet's type theorems for non linear singular partial differential equations - Maillet's type theorems for non linear singular partial differential equations without linear part - Holomorphic and singular solutions of non linear singular partial differential equations - On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations - Maillet's type theorems for non linear singular integro-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.
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Autorenporträt
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.
Inhaltsangabe
- Operators with regular singularities. One variable case
- Operators with regular singularities. Several variables case
- Formal and convergent solutions of singular partial differential equations
- Local study of differential equations of the form xy' f(x,y) near x 0
- Holomorphic and singular solutions of non linear singular first order partial differential equations
- Maillet's type theorems for non linear singular partial differential equations
- Maillet's type theorems for non linear singular partial differential equations without linear part
- Holomorphic and singular solutions of non linear singular partial differential equations
- On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations
- Maillet's type theorems for non linear singular integro-differential equations
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 '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 integro-differential 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.
- Operators with regular singularities. One variable case
- Operators with regular singularities. Several variables case
- Formal and convergent solutions of singular partial differential equations
- Local study of differential equations of the form xy' f(x,y) near x 0
- Holomorphic and singular solutions of non linear singular first order partial differential equations
- Maillet's type theorems for non linear singular partial differential equations
- Maillet's type theorems for non linear singular partial differential equations without linear part
- Holomorphic and singular solutions of non linear singular partial differential equations
- On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations
- Maillet's type theorems for non linear singular integro-differential equations
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 '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 integro-differential 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.
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