This book is concerned with functional methods (nonlinear semigroups of contractions, nonlinear m-accretive operators and variational techniques) in the theory of nonlinear partial differential equations of elliptic and parabolic type. In particular, applications to the existence theory of nonlinear parabolic equations, nonlinear Fokker-Planck equations, phase transition and free boundary problems are presented in details. Emphasis is put on functional methods in partial differential equations (PDE) and less on specific results.
Author(s): Viorel Barbu
Edition: 1
Publisher: World Scientific
Year: 2022
Language: English
Pages: 220
Tags: Banach Spaces, Sobolev Spaces, Monotone Operators, Accretive Operators, Nonlinear PDEs
Contents
Preface
1. Preliminaries
1.1 Geometric Properties of Banach Spaces
1.2 Convex Functions
1.3 Sobolev Spaces
1.4 Infinite-dimensional Sobolev Spaces
2. Monotone and Accretive Operators in Banach Spaces
2.1 Maximal Monotone Operators
2.2 Accretive Operators
2.3 Existence Theory for the Cauchy Problem
3. Nonlinear Elliptic Boundary Value Problems
3.1 Nonlinear Elliptic Problems of Divergence Type
3.2 Semilinear Elliptic Operators in Lp(Ω)
3.3 Quasilinear Partial Differential Equations of First Order
3.4 Porous Media Equations with Drift Term
4. Nonlinear Dissipative Dynamics
4.1 Semilinear Parabolic Equations
4.2 The Porous Media Equation
4.3 The Phase Field Transition with Mushy Region
4.4 The Self-organized Criticality Equation
4.5 The Conservation Law Equation
4.6 The Fokker–Planck Equation
4.7 A Fokker–Planck Like Parabolic Equation in RN
4.8 Generalized Fokker–Planck Equation
Bibliography
Index
