This graduate-level book is an introduction to the modern theory of partial differential equations (PDEs) with an emphasis on elliptic PDEs. It includes the most important and commonly used methods and central results for elliptic PDEs.
A number of great strategies for solving PDEs analytically and studying them qualitatively are discussed. The Galerkin, semigroup, variational, and topological methods are utilized for existence problems. Some aspects of Brownian motion are also presented. The existence of weak solutions and verification of the regularity of weak solutions are considered as guiding principles in the book.
With regard to the diffusion method, solutions of elliptic equations are obtained as asymptotic equilibria of the parabolic PDEs. Other methods presented in the book for the existence of solutions of elliptic PDEs are the difference method, the Perron method, and the alternating method of H.A. Schwarz based on the maximum principle. Several versions of maximum principle that can be used in nonlinear PDEs are also included.
For the variational method, the theory of Sobolev spaces is developed. For the regularity of weak solutions obtained by using the variational method, most of the fundamental embedding theorems of Sobolev, Morrey, and John-Nirenberg are also presented. Schauder regularity theory for solutions in Hӧlder spaces and solvability by a basic continuity method are also presented. An entire chapter is devoted to the Moser iteration technique which is used to extend the properties known for harmonic functions to those for solutions of general elliptic PDEs. It is shown in detail how the Harnack inequality can be used for the regularity of solutions. The book also develops methods that are useful for the study of nonlinear PDEs that appear in sciences, engineering and economics.
There is a summary of each chapter at its end and then are exercises which are to complement and expand the depth of the content in the chapter. One important feature of the book is that relations between different types of PDEs, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, and viscosity solutions for elliptic PDEs, are discussed. The reader will find a smooth flow of the development of the theory of PDEs and various solution methods for multitude of problems in sciences, engineering and economics.
The book is undoubtedly a success in the presentation of diverse methods in PDEs at such an introductory level. The reader has a great opportunity to learn basic techniques underlying current research in elliptic PDEs and be motivated for advanced theory of more general elliptic PDEs and nonlinear PDEs.
Dhruba Adhikari is an assistant professor of mathematics at Southern Polytechnic State University, Marietta, Georgia.
Introduction: What are Partial Differential Equations?
1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order
2 The Maximum Principle
3 Existence Techniques I: Methods Based on the Maximum Principle
4 Existence Techniques II: Parabolic Methods. The Heat Equation
5 Reaction-Diffusion Equations and Systems
6 Hyperbolic Equations
7 The Heat Equation, Semigroups, and Brownian Motion
8 Relationships between Different Partial Differential Equations
9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III)
10 Sobolev Spaces and \(L^2\) Regularity theory
11 Strong solutions
12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV)
13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash
Appendix: Banach and Hilbert spaces. The \(L^p\) Spaces
Index of Notation