Dual Variational Approach to Nonlinear Diffusion Equations

Description

This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well. Preface Introduction Contents Acronyms 1 Nonlinear Diffusion Equations with Slow and Fast Diffusion 1.1 A Continuous Potential with a Polynomial Growth 1.1.1 Functional Framework and Preliminaries 1.1.2 The Minimization Problem 1.1.3 Main Results for the Slow Diffusion Case 1.2 A Free Boundary Problem for a Special Fast Diffusion Case 2 Weakly Coercive Nonlinear Diffusion Equations 2.1 Problem Presentation 2.2 Functional Setting and the Minimization Problem 2.3 A Time and Space Dependent Potential 2.4 A Time Dependent Potential 3 Nonlinear Diffusion Equations with a Noncoercive Potential 3.1 Problem Statement and the Functional Framework 3.2 The Minimization Problem 3.3 Main Results for the Noncoercive Case 3.4 Existence in a Self-organized Criticality (SOC) Model 4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary Conditions 4.1 Problem Presentation 4.2 The Strongly Coercive Case 4.3 The Weakly Coercive Case 4.4 The Semigroup Approach 4.4.1 The Obstacle Problem 4.4.2 The Total Variation Wentzell Flow 5 A Nonlinear Control Problem in Image Denoising 5.1 Problem Presentation and Preliminaries 5.2 Existence for the Minimization Problem 5.3 Approximating Problem 5.3.1 Optimality Conditions for the Approximating Problem 5.3.2 Numerical Algorithm 5.4 A Potential with a Polynomial Growth 6 An Optimal Control Problem for a Phase Transition Model 6.1 Presentation of the Problem 6.2 Existence in the State System and Control Problem 6.3 Approximating Control Problem (P) 6.4 Optimality Conditions 6.4.1 The Second Approximating Control Problem (P,σ) 6.4.2 Optimality Conditions for (P,σ) 6.4.3 Optimality Conditions for (P) 7 Appendix 7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces 7.2 Operators in Banach Spaces 7.3 Convex Functions and Subdifferential Mappings References Index