Elliptic Partial Differential Equations of Second Order
Description
From the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student." --New Zealand Mathematical Society, 1985
Reviews / Votes
From the reviews:"The aim of the book is to present "the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process". The book is divided into two parts. The first (Chapters 2-8) is devoted to the linear theory, the second (Chapters 9-15) to the theory of quasilinear partial differential equations. These 14 chapters are preceded by an Introduction (Chapter 1) which expounds the main ideas and can serve as a guide to the book. ...The authors have succeeded admirably in their aims; the book is a real pleasure to read".
Mathematical Reviews,1986
"Advanced students and professionals are snapping up this paperback text on linear and quasilinear partial differential equations. Whether you use their book as textbook or reference, the authors give you plenty to think about and work on, including an epilogue summarizing the latest research."
Amazon.com delivers Mathematics and Statistics e-bulletin, July 2001
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Persons
Biography of David Gilbarg
David Gilbarg was born in New York in 1918, and was educated there through udergraduate school. He received his Ph.D. degree at Indiana University in 1941. His work in fluid dynamics during the war years motivated much of his later research on flows with free boundaries. He was on the Mathematics faculty at Indiana University from 1946 to 1957 and at Stanford University from 1957 on. His principal interests and contributions have been in mathematical fluid dynamics and the theory of elliptic partial differential equations.
Biography of Neil S. Trudinger
Neil S. Trudinger was born in Ballarat, Australia in 1942. After schooling and undergraduate education in Australia, he completed his PhD at Stanford University, USA in 1966. He has been a Professor of Mathematics at the Australian National University, Canberra since 1973. His research contributions, while largely focussed on non-linear elliptic partial differential equations, have also spread into geometry, functional analysis and computational mathematics. Among honours received are Fellowships of the Australian Academy of Science and of the Royal Society of London.
Content
Chapter 2. Laplace's Equation
2.1 The Mean Value Inequalities
2.2 Maximum and Minimum Principle
2.3 The Harnack Inequality
2.4 Green's Representation
2.5 The Poisson Integral
2.6 Convergence Theorems
2.7 Interior Estimates of Derivatives
2.8 The Dirichlet Problem; the Method of Subharmonic Functions
2.9 Capacity
Problems
Chapter 3. The Classical Maximum Principle
3.1 The Weak Maximum Principle
3.2 The Strong Maximum Principle
3.3 Apriori Bounds
3.4 Gradient Estimates for Poisson's Equation
3.5 A Harnack Inequality
3.6 Operators in Divergence Form
Notes
Problems
Chapter 4. Poisson's Equation and Newtonian Potential
4.1 Hölder Continuity
4.2 The Dirichlet Problem for Poisson's Equation
4.3 Hölder Estimates for the Second Derivatives
4.4 Estimates at the Boundary
4.5 Hölder Estimates for the First Derivatives
Notes
Problems
Chapter 5. Banach and Hilbert Spaces
5.1 The Contraction Mapping
5.2 The Method of Cintinuity
5.3 The Fredholm Alternative
5.4 Dual Spaces and Adjoints
5.5 Hilbert Spaces
5.6 The Projection Theorem
5.7 The Riesz Representation Theorem
5.8 The Lax-Milgram Theorem
5.9 The Fredholm Alternative in Hilbert Spaces
5.10 Weak Compactness
Notes
Problems
Chapter 6. Classical Solutions; the Schauder Approach
6.1 The Schauder Interior Estimates
6.2 Boundary and Global Estimates
6.3 The Dirichlet Problem
6.4 Interior and Boundary Regularity
6.5 An Alternative Approach
6.6 Non-Uniformly Elliptic Equations
6.7 Other Boundary Conditions; the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities
6.9 Appendix 2: Extension Lemmas
Notes
Problems
Chapter 7. Sobolev Spaces
7.1 L^p spaces
7.2 Regularization and Approximation by Smooth Functions
7.3 Weak Derivatives
7.4 The Chain Rule
7.5 The W^(k,p) Spaces
7.6 DensityTheorems
7.7 Imbedding Theorems
7.8 Potential Estimates and Imbedding Theorems
7.9 The Morrey and John-Nirenberg Estimes
7.10 Compactness Results
7.11 Difference Quotients
7.12 Extension and Interpolation
Notes
Problems
Chapter 8 Generalized Solutions and Regularity
8.1 The Weak Maximum Principle
8.2 Solvability of the Dirichlet Problem
8.3 Diferentiability of Weak Solutions
8.4 Global Regularity
8.5 Global Boundedness of Weak Solutions
8.6 Local Properties of Weak Solutions
8.7 The Strong Maximum Principle
8.8 The Harnack Inequality
8.9 Hölder Continuity
8.10 Local Estimates at the Boundary
8.11 Hölder Estimates for the First Derivatives
8.12 The Eigenvalue Problem
Notes
Problems
Chapter 9. Strong Solutions
9.1 Maximum Princiles for Strong Solutions
9.2 L^p Estimates: Preliminary Analysis
9.3 The Marcinkiewicz Interpolation Theorem
9.4 The Calderon-Zygmund Inequality
9.5 L^p Estimates
9.6 The Dirichlet Problem
9.7 A Local Maximum Principle
9.8 Hölder and Harnack Estimates
9.9 Local Estimates at the Boundary
Notes
Problems
Part II: Quasilinear Equations
Chapter 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
Problems
Chapter 11. Topological Fixed Point Theorems and Their Application
11.1 The Schauder Fixes Point Theorem
11.2 The Leray-Schauder Theorem: a Special Case
11.3 An Application
11.4 The Leray-Schauder Fixed Point Theorem
11.5 Variational Problems
Notes
Chapter 12. Equations in Two Variables
12.1 Quasiconformal Mappings
12.2 hölder Gradient Estimates for Linear Equations
12.3 The Dirichlet Problem for Uniformly Elliptic Equations
12.4 Non-Uniformly Elliptic Equations
Notes
Problems
Chapter 13. Hölder Estimates for