Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators, while demonstrating how the theory of semigroups represents a powerful tool to analyze general parabolic equations.
- Useful for students and researchers as an introduction to the field of partial differential equations of elliptic and parabolic types
- Introduces the reader to the theory of operator semigroups as a tool for the analysis of partial differential equations
Table of Contents
1. Function spaces. I. Semigroups of bounded operators. 2. Strongly continuous semigroups. 3. Analytic semigroups. II. Parabolic equations. 4. Elliptic and parabolic maximum principles. 5 Prelude to parabolic equations: the heat equation and the Gauss-Weierstrass semigroup in Cᵇ(Rd). 6. Parabolic equations in Rd. 7. Parabolic equations in Rd + with Dirichlet boundary conditions. 8. Parabolic equations in Rd+ with more general boundary conditions. 9 Parabolic equations in bounded smooth domains Ω. III Elliptic equations. 10. Elliptic equations in Rͩ. 11. Elliptic equations in Rd+ with homogeneous Dirichlet boundary conditions. 12. Elliptic equation in Rd+ with general boundary conditions. 13 Elliptic equations on smooth domains Ω. 14 Elliptic operators and analytic semigroups. 15. Kernel estimates. IV. Appendices. A Basic notion of Functional Analysis in Banach spaces. B Smooth domains and extension operators.
Luca Lorenzi is Full Professor of Mathematical Analysis at University of Parma (Italy). He received his PhD in Mathematics from the University of Pisa (Italy) in 2001. In 2000 he got a permanent position as assistant professor in Mathematical Analysis at the University of Parma and he was promoted to associate professor in 2006 still at the University of Parma. In 2013 he got the Italian National habilitation as full professor in Mathematical Analysis. His research interests are mainly focused on Partial Differential Equations. In particular, Evolution Equations and Operator Semigroups, Non-autonomous Differential Equations, Elliptic and Parabolic Differential Operators with Unbounded Coefficients. He has authored or co-authored over 60 papers published on international journals and three scientific monographs.
Abdelaziz Rhandi is Full Professor of Mathematical Analysis at University of Salerno (Italy). Before joining Salerno University in 2006, he served as Full Professor of Mathematics at the University of Marrakesh (Morocco) since 1999. He received his first PhD in Mathematics from the University of Besançon (France) and the second one from the University of Tübingen (Germany). In 2003, he got the Alexander von Humboldt Fellow and was the winner of the 2006 HP Technology for Teaching Higher Education Prize. He has worked as a visiting professor in several universities in Algeria, France, Germany, Italy, Tunisia and USA. His research interests are mainly focused on Applied Functional Analysis and Partial Differential Equations. In particular, Evolution Equations and Operator Semigroups, Non-autonomous Differential Equations, Elliptic and Parabolic Differential Operators with Unbounded Coefficients.