Form Methods for Evolution Equations

This book is devoted to the study of evolution equations via form methods. The theory is presented both in the language of Kato, with densely defined forms, and that of Lions, in the spirit of Gelfand triples. The main object is the semigroup associated with a form, for which topics such as positivity, invariance of closed convex sets and the Trotter product formula are discussed. A wide range of applications is treated, for example, parabolic equations with various boundary conditions, the Stokes operator, the Dirichlet-to-Neumann operator, and non-autonomous semilinear parabolic equations. The book grew out of the Internet Seminar “Form Methods for Evolution Equations, and Applications” organized by the authors. Each of the 19 chapters is devoted to one particular subject and includes exercises. As a special feature, carefully placed “interludes” provide background results, all with complete proofs, to make the book fully self-contained.