Form Methods for Evolution Equations: (316 Operator Theory: Advances and Applications)

About the Book

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.

Table of Contents:

Chapter 1. C₀-semigroups.- Chapter 2. Characterisation of generators of C₀-semigroups.- Chapter 3. Holomorphic semigroups.- Chapter 4. The Sobolev space H¹ and applications.- Chapter 5. Forms and operators.- Chapter 6. Adjoint operators and compactness.- Chapter 7. Neumann and Robin boundary conditions.- Chapter 8. The Dirichlet-to-Neumann operator.- Chapter 9. Invariance of closed convex sets.- Chapter 10. Interpolation of holomorphic semigroups.- Chapter 11. Elliptic operators.- Chapter 12. Sectorial forms.- Chapter 13. Approximation of strongly continuous semigroups.- Chapter 14. Form convergence theorems.- Chapter 15. The Trotter product formula for forms.- Chapter 16. The Stokes operator.- Chapter 17. Non-autonomous equations.- Chapter 18. Maximal regularity for non-autonomous equations.- Chapter 19. Nonlinear non-autonomous equations.

About the Author :
Wolfgang Arendt is a senior professor at Ulm University. He studied mathematics and physics in Berlin, Nice, and Tübingen, earning his doctorate under H.H. Schaefer in 1979, and completing his habilitation in 1985. From 1987 to 1995, he was a professor at the Université de Franche-Comté in Besançon, before joining Ulm University as the chair of applied analysis (1995-2018). His research focuses on functional analysis, operator semigroups, and partial differential equations. Wolfgang Arendt is the co-author of "One-Parameter Semigroups of Positive Operators" (Springer LNM 1184, second ed. 2026), "Vector-Valued Laplace Transforms and Cauchy Problems"(Birkhäuser 2011), and "Partial Differential Equations" (Springer 2023). He has supervised numerous doctoral students, serves as the Editor-in-Chief of the Journal of Evolution Equations, and has co-founded the International Internet Seminar on Evolution Equations. Hendrik Vogt is a lecturer at the University of Bremen. He studied mathematics at the University of Oldenburg and earned his doctorate under J. Voigt from TU Dresden in 2001. From 2002 to 2003, he was a visiting assistant professor at the University of Memphis, before returning to Dresden to complete his habilitation. From 2010 to 2014, Hendrik Vogt was a deputy professor at the technical universities in Clausthal, Hamburg Harburg, and Dresden. Hendrik Vogt's research focuses on functional analysis, with applications to operator semigroups, spectral theory, and partial differential equations. His work has led to numerous publications in international peer-reviewed journals. In addition to his research, Hendrik Vogt has extensive teaching experience across core areas of higher mathematics, ranging from introductory analysis to advanced topics such as spectral theory and evolution equations. Jürgen Voigt studied mathematics in Stuttgart, Tübingen, Paris and Munich. After earning his PhD under J. Weidmann and completing his habilitation at Ludwig Maximilian University in Munich, he was appointed a professor at the University of Oldenburg. From 1993, he held the chair of applied functional analysis at Dresden University of Technology, where he has continued to participate in research as a professor emeritus since 2010. Jürgen Voigt's research has had a significant impact on the theory of one-parameter semigroups and evolution equations and his contributions to Schrödinger operators and transport theory are well-known. He is the author of the book "A Course on Topological Vector Spaces", which appeared in the Compact Textbooks series published by Birkhäuser. Right from its start in 1997, he was an enthusiastic scientific advisor and participant of the annual Internet Seminar on Evolution Equations and Operator Semigroups.