Includes bibliographical references (p. [373]-380) and index.
Contents:
Ch. 1. Quasi-linear equations and the Cauchy-Kowalewski theorem
Ch. 2. The Laplace equation
Ch. 3. The double layer potential and boundary value problems
Ch. 4. Integral equations and Eigenvalue problems
Ch. 5. The heat equation
Ch. 6. The wave equation
Ch. 7. Quasi-linear equations of first order
Ch. 8. Non-linear equations of first-order
Ch. 9. Linear elliptic equations with measurable coefficients
10. DeGiorgi classes.
Summary:
"This self-contained text offers an elementary introduction to partial differential equations (pdes), primarily focusing on linear equations, but also providing some perspective on nonlinear equations. The classical treatment is mathematically rigorous with a generally theoretical layout, though indications to some of the physical origins of pdes are made throughout in references to potential theory, similarity solutions for the porous medium equation, generalized Riemann problems, and others. The material begins with a focus on the Cauchy-Kowalewski theorem, discussing the notion of characteristic surfaces to classify pdes. Next, the Laplace equation and connected elliptic theory are treated, as well as integral equations and solutions to eigenvalue problems. The heat equation and related parabolic theory are then presented, followed by the wave equation in its basic aspects. An introduction to conservation laws, the uniqueness theorem, viscosity solutions, ill-posed problems, and nonlinear equations of first order round out the key subject matter. Large parts of this revised second edition have been streamlined and rewritten to incorporate years of classroom feedback, correct errors, and improve clarity. Most of the necessary background material has been incorporated into the complements and certain nonessential topics have been given reduced attention (noticeably, numerical methods) to improve the flow of presentation. The exposition is replete with examples, problems and solutions that complement the material to enhance understanding and solidify comprehension. The only prerequisites are advanced differential calculus and some basic Lp theory. The work can serve as a text for advanced undergraduates and graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference for applied mathematicians and mathematical physicists."--Publisher's description.