1. Introduction.- I. Linear Equations.- 2. Laplace’s Equation.- 3. The Classical Maximum Principle.- 4. Poisson’s Equation and the Newtonian Potential.- 5. Banach and Hubert Spaces.- 6. Classical Solutions; the Schauder Approach.- 7. Sobolev Spaces.- 8. Generalized Solutions and Regularity.- 9. Strong Solutions.- II. Quasilinear Equations.- 10. Maximum and Comparison Principles.- 11. Topological Fixed Point Theorems and Their Application.- 12. Equations in Two Variables.- 13. Hölder Estimates for the Gradient.- 14. Boundary Gradient Estimates.- 15. Global and Interior Gradient Bounds.- 16. Equations of Mean Curvature Type.- 17. Fully Nonlinear Equations.- Epilogue.- Notation Index.
Springer Book Archives
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.
“This book is a bibliographical monument to the theory of both
theoretical and applied PDEs that has not acquired any flaws due to
its age. On the contrary, it remains a crucial and essential tool
for the active research in the field.” (Francesco Petitta, SIAM
Review, Vol. 61 (4), December, 2019)
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."
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2001
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