This textbook, the author of which is a well-regarded specialist in PDEs and nonlinear analysis, is a revised version of a book which appeared in Romanian in 1993. It concerns the fundamental topics of second order PDEs, demonstrated by the examples of the Poisson equation, the heat equation and the wave equation. These topics are:
– classical solutions to boundary value problems for elliptic equations (based on potential theory, boundary integral equations and Riesz-Schauder theory as well as on the maximum principle and the concept of subharmonic functions);
– weak solutions of such boundary value problems in Sobolev spaces (based on the Lax-Milgram lemma), interior and boundary regularity, generalizations to nonlinear equations (monotonicity methods), eigenvalue problems;
– weak solutions to initial boundary value problems for parabolic and hyperbolic equations (based on the Fourier-method);
– semigroup approach to such initial boundary value problems (based on the Hille-Yosida theorem);
– weak and classical solutions to Cauchy problems for parabolic and hyperbolic equations.
The book is very careful written and can perfectly serve as a graduate-level course at universities. It presents detailed and pedagogic proofs. Each chapter includes well-selected excercises with hints.