This book concerns the mathematical analysis of quasilinear partial differential equations (PDEs) where the leading operator is nonlinear, elliptic and in divergence form. Techniques such as monotone operators, pseudomonotone operators, accretive operators, potential operators, variational inequalities and set-valued mappings form the cornerstone basis for the presented analysis. Special attention is also devoted to penalty methods. The author treats both steady-state problems in part I and corresponding evolutionary problems in part II. For time-dependent problems, the Rothe and Faedo-Galerkin methods are incorporated in detail.
Each section has the same structure: a general abstract framework is accompanied by applications of theoretical results to carefully selected examples starting from sample cases up to the cases that have their origin in the physical sciences (thermofluid mechanics, thermoelasticity, reaction-diffusion problems, material science). Frequently the author shows that dealing with a specific (system of) PDE(s) can strengthen the results obtained by abstract methods. Each section is completed with exercises and a representative list of relevant literature. This carefully written book, addressed to graduate and PhD students and researchers in PDEs, applied analysis and mathematical modelling contains on one hand several mathematical approaches developed to understand the basic mathematical properties of nonlinear PDEs and on the other hand many interesting examples. Some are solved, others are complemented by hints and the rest are left for the reader to complete.