Partial Differential Equations in Action

This work is an updated version of a book evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In the second part, chapters 6 to 10 concentrate on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems, while Chapter 11 deals with vector-valued conservation laws, extending the theory developed in Chapter 4. The main differences with respect to the previous editions are: a new section on reaction diffusion models for population dynamics in a heterogeneous environment; several new exercises in almost all chapters; a general restyling and a reordering of the last chapters. The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering.

1 Introduction. - 2 Diffusion. - 3 The Laplace Equation. - 4 Scalar Conservation Laws and First Order Equations. - 5 Waves and Vibration. - 6 Elements of Functional Analysis. - 7 Distributions and Sobolev Spaces. - 8 Variational Formulation of Elliptic Problems. - 9 Weak Formulation of Evolution Problems. - 10 More Advanced Topics. - 11 Systems of Conservation Laws. - Appendix A: Measures and Integrals. - Appendix B: Identities and Formulas.