A survey of topics related to partial differential equations

The book has grown out of readings of current papers during the past ten years. It is not a systematic treatise on the theory of partial differential equations. It contains short accounts (sections) of the elements of many sides of the theory not usually combined in a single volume. The main sides which are dealt with are existence, uniqueness, regularity of solutions to linear and nonlinear, stationary or evolution equations, qualitative properties of solutions (resonances, attractors, inertial manifolds), dynamical systems. Related Harnack, Moser-Trudinger, Sobolev-Poincaré, Strichartz inequalities, and function spaces such as BMO, BV, Morrey, Orlicz, Sobolev spaces, Kato classes of functions, are investigated. Sections include historical comments, definitions, main results without detailed proofs, examples and applications extracted from current works. This book has been written for young researchers in mathematics and applied sciences (dynamics, chemistry, biology). With sections presented in an alphabetical order, an author and subject indexes, it can be a useful tool for advanced students entering in the field. A solid undergraduate background in mathematics is required.