Linear and Semilinear Partial Differential Equations: An Introduction

The text is intended for students who wish a concise and rapid introduction to some main topics in PDEs, necessary for understanding current research, especially in nonlinear PDEs. Organized on three parts, the book guides the reader from fundamental classical results, to some aspects of the modern theory and furthermore, to some techniques of nonlinear analysis. Compared to other introductory books in PDEs, this work clearly explains the transition from classical to generalized solutions and the natural way in which Sobolev spaces appear as completions of spaces of continuously differentiable functions with respect to energetic norms. Also, special attention is paid to the investigation of the solution operators associated to elliptic, parabolic and hyperbolic non-homogeneous equations anticipating the operator approach of nonlinear boundary value problems. Thus the reader is made to understand the role of linear theory for the analysis of nonlinear problems. Organizes the subject on three levels: classical, modern and semi-linear theory Clearly explains the transition from classical to generalized solutions Introduces, in the beginning, the Sobolev spaces as completions of spaces of continuously differentiable functions with respect to energetic norms Presents the solution operators associated to non-homogeneous equations and anticipates the operator method for nonlinear problems Covers most of the main topics usually studied in standard courses Provides a rigorous theoretical treatment by organizing the material around theorems and proofs Includes numerous exercises and problems to assimilate and extend the theory Provides a solid base for further study and inspiration for future research in the field Provides material for three one-semester courses: a beginner, an advanced and a master course