Elliptic partial differential equations of second order /
David Gilbarg, Neil S. Trudinger.
- Primera edición 1983, reimpresión año 2001
- xiii, 517 páginas
- Classics in mathematics .
Originally published as vol. 224 of the grundlehren der mathematischen Wissenschaften.
Chapter 1. Introduction Part I: Linear EquationsChapter 2. Laplace's Equation2.1 The Mean Value Inequalities2.2 Maximum and Minimum Principle2.3 The Harnack Inequality2.4 Green's Representation2.5 The Poisson Integral2.6 Convergence Theorems2.7 Interior Estimates of Derivatives2.8 The Dirichlet Problem; the Method of Subharmonic Functions2.9 CapacityProblemsChapter 3. The Classical Maximum Principle3.1 The Weak Maximum Principle3.2 The Strong Maximum Principle3.3 Apriori Bounds3.4 Gradient Estimates for Poisson's Equation3.5 A Harnack Inequality3.6 Operators in Divergence FormNotesProblemsChapter 4. Poisson's Equation and Newtonian Potential4.1 Hölder Continuity4.2 The Dirichlet Problem for Poisson's Equation4.3 Hölder Estimates for the Second Derivatives4.4 Estimates at the Boundary4.5 Hölder Estimates for the First DerivativesNotes ProblemsChapter 5. Banach and Hilbert Spaces5.1 The Contraction Mapping5.2 The Method of Cintinuity5.3 The Fredholm Alternative5.4 Dual Spaces and Adjoints5.5 Hilbert Spaces5.6 The Projection Theorem5.7 The Riesz Representation Theorem5.8 The Lax-Milgram Theorem5.9 The Fredholm Alternative in Hilbert Spaces5.10 Weak CompactnessNotesProblemsChapter 6. Classical Solutions; the Schauder Approach6.1 The Schauder Interior Estimates6.2 Boundary and Global Estimates6.3 The Dirichlet Problem6.4 Interior and Boundary Regularity6.5 An Alternative Approach6.6 Non-Uniformly Elliptic Equations6.7 Other Boundary Conditions; the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities6.9 Appendix 2: Extension LemmasNotesProblemsChapter 7. Sobolev Spaces7.1 L^p spaces7.2 Regularization and Approximation by Smooth Functions7.3 Weak Derivatives7.4 The Chain Rule7.5 The W^(k,p) Spaces7.6 Density Theorems7.7 Imbedding Theorems7.8 Potential Estimates and Imbedding Theorems7.9 The Morrey and John-Nirenberg Estimes7.10 Compactness Results7.11 Difference Quotients7.12 Extension and InterpolationNotesProblemsChapter 8 Generalized Solutions and Regularity8.1 The Weak Maximum Principle8.2 Solvability of the Dirichlet Problem8.3 Diferentiability of Weak Solutions8.4 Global Regularity8.5 Global Boundedness of Weak Solutions8.6 Local Properties of Weak Solutions8.7 The Strong Maximum Principle8.8 The Harnack Inequality8.9 Hölder Continuity8.10 Local Estimates at the Boundary8.11 Hölder Estimates for the First Derivatives8.12 The Eigenvalue ProblemNotesProblemsChapter 9. Strong Solutions9.1 Maximum Princiles for Strong Solutions9.2 L^p Estimates: Preliminary Analysis9.3 The Marcinkiewicz Interpolation Theorem9.4 The Calderon-Zygmund Inequality9.5 L^p Estimates9.6 The Dirichlet Problem9.7 A Local Maximum Principle9.8 Hölder and Harnack Estimates9.9 Local Estimates at the BoundaryNotesProblemsPart II: Quasilinear EquationsChapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes ProblemsChapter 11. Topological Fixed Point Theorems and Their Application11.1 The Schauder Fixes Point Theorem11.2 The Leray-Schauder Theorem: a Special Case11.3 An Application11.4 The Leray-Schauder Fixed Point Theorem11.5 Variational ProblemsNotesChapter 12. Equations in Two Variables12.1 Quasiconformal Mappings12.2 hölder Gradient Estimates for Linear Equations12.3 The Dirichlet Problem for Uniformly Elliptic Equations12.4 Non-Uniformly Elliptic EquationsNotesProblemsChapter 13. Hölder Estimates for the Gradient13.1 Equations of Divergence Form13.2 Equations in Two Variables13.3 Equations of General Form; the Interior Estimate13.4 Equations of General Form; the Boundary Estimate13.5 Application to the Dirichlet ProblemNotesChapter 14. Boundary Gradient Estimates14.1 General Domains14.2 Convex Domains14.3 Boundary Curvature Conditions14.4 Non-Existence Results14.5 Continuity Estimates14.6 Appendix: Boundary Curvature and the Distance FunctionNotesProblemsChapter 15. Global and Interior Gradient Bounds15.1 A Maximum Principle for the Gradient15.2 The General Case15.3 Interior Gradient Bounds15.4 Equations in Divergence Form15.5 Selected Existence Theorems15.6 Existence Theorems for Continuous Boundary ValuesNotesProblemsChapter 16. Equations of Mean Curvature Type16.1 Hypersurfaces in R^(n+1) 16.2 Interior Gradient Bounds16.3 Application to the Dirichlet Problem16.4 Equations in Two Independent Variables16.5 Quasiconformal Mappings16.6 Graphs with Quasiconformal Gauss Map16.7 Applications to Equations of mean Curvature Type16.8 Appendix Elliptic Parametric FunctionalsNotesProblemsChapter 17. Fully Nonlinear Equations17.1 Maximum and Comparison Principles17.2 The Method of Continuity17.3 Equations in Two Variables17.4 Hölder Estimates for Second Derivatives17.5 Dirichlet Problem for Uniformly Elliptic Equations17.6 Second Derivative Estimates for Equations of Monge-Ampère Type17.7 Dirichlet Problem for Equations of Monge-Amperère Type17.8 Global Second Derivative Hölder Estimates17.9 Nonlinear Boundary Value ProblemsNotesProblemsBibliographyEpilogueSubject IndexNotation Index
COBERTURA BIBLIOGRÁFICA (Departamento de Matemática) Asignatura: Ecuaciones Diferenciales Parciales