Partial Differential Equations
The book is intended for students wishing to find an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view toward nonlinear problems. It includes maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. The book also develops the main methods for obtaining estimates for solutions of elliptic equations; Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. The book can be used for a one-year course on partial differential equations. Having some experience in teaching PDEs, I am always curious to see a new textbook in the field. I have found the book by J. Jost very well written. The concentration on elliptic equations creates a new possibility for an exposition of the main features of evolution equations. Both the "Preface" and the "Introduction" are rather helpful - it could be useful for the reader to come back to them from time to time to put the ideas together. I share the author’s opinion that his book helps "in guiding the reader through an area of mathematics that does not allow a unified structural approach, but rather derives its fascination from the multitude and diversity of approaches and methods..."