Partial Differential Equations

Jürgen Jost is currently a codirector of the Max Planck Institute for Mathematics in the Sciences and an honorary professor of mathematics at the University of Leipzig.

• Preface.- Introduction: What are Partial Differential Equations?.- 1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- 2 The Maximum Principle.- 3 Existence Techniques I: Methods Based on the Maximum Principle.- 4 Existence Techniques II: Parabolic Methods. The Heat Equation.- 5 Reaction-Diffusion Equations and Systems.- 6 Hyperbolic Equations.- 7 The Heat Equation, Semigroups, and Brownian Motion.- 8 Relationships between Different Partial Differential Equations.- 9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III).- 10 Sobolev Spaces and L^2 Regularity theory.- 11 Strong solutions.- 12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- 13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.- Appendix: Banach and Hilbert spaces. The L^p-Spaces.- References.- Index of Notation.- Index.

From the book reviews: "This graduate-level book is an introduction to the modern theory of partial differential equations (PDEs) with an emphasis on elliptic PDEs. ... The book is undoubtedly a success in the presentation of diverse methods in PDEs at such an introductory level. The reader has a great opportunity to learn basic techniques underlying current research in elliptic PDEs and be motivated for advanced theory of more general elliptic PDEs and nonlinear PDEs." (Dhruba Adhikari, MAA Reviews, December, 2014) "This revised version gives an introduction to the theory of partial differential equations. ... Every chapter has at the end a very helpful summary and some exercises. This book is very useful for a PhD course." (Vincenzo Vespri, Zentralblatt MATH, Vol. 1259, 2013) "Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations." - Alain Brillard, Mathematical Reviews "Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics." - Nick Lord, The Mathematical Gazette