Applied Functional Analysis and Partial Differential Equations
This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools that PDEs require. The contents of the book is organized in such a way that an undergraduate student, with a basic knowledge of calculus, Lebesgue integrals and the Cauchy theorem of complex analysis, can read it without problems. The other material in the book is self-contained, giving a very nice introduction to functional-analytical tools needed to study the theory of PDEs. The basics of linear operators in Banach and Hilbert spaces are given at the beginning of the book. Then, Sobolev spaces are introduced together with basic properties of distributions and Fourier transforms. Elliptic problems are studied together with the question of regularity of weak solutions to these. In the last three chapters, the reader is led through the theory of semigroups of linear operators, weakly nonlinear elliptic problems and semilinear parabolic equations. The finite difference method and the method of Galerkin approximations (for both parabolic and wave equations) are given as examples. The chapter on semilinear parabolic equations includes studies of the stability of fluid flows and, more generally, of the dynamics generated by dissipative systems, numerical PDEs, elliptic and hyperbolic PDEs, and quantum mechanics. The book is nicely written and is self-contained. Therefore it can be recommended as one of the basic readings for the study of PDEs.