Introduction to classical and variational partial differential equations - Cioranescu, Doina (University of the Philippines Press, 2012)
The study of partial differential equations is at the crossroads of mathematical analysis, measure theory, topology, differential geometry, scientific computing, and many other branches of mathematics. Modeling physical phenomena, partial differential equations are fascinating topics because of their increasing presence in treating real physical processes. In recent years, PDEs have become essential modeling tools in fields such as materials science, mathematical finance, quantum mechanics, biology and biomedicine, and environmental sciences. The aim of this book is to introduce classical and variational PDEs to graduate and post-graduate students in Mathematics. It concerns mainly second order linear partial differential equations and consists of two parts. Part I gives a comprehensive overview of classical PDEs, that is, equations which admit smooth (strong) solutions, verifying the equations pointwise. Classical solutions of the Laplace, heat, and wave equations are given. Part II deals with variational PDEs, where weak solutions are considered. These solutions verify a weak formulation of the equations and belong to suitable spaces of functions, the Sobolev spaces. The theory of Sobolev spaces provides the foundation for the study of variational PDEs. A comprehensive and detailed presentation of these spaces and the Sobolev embeddings is presented. Examples of variational elliptic, parabolic, and hyperbolic problems with different boundary conditions are also discussed.