Books in the Progress in Nonlinear Differential Equations and Their Applications series

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

This monograph explores the concept of the Brouwer degree and its continuing impact on the development of important areas of nonlinear analysis.

This monograph develops a framework for time-optimal control problems, focusing on minimal and maximal time-optimal controls for linear-controlled evolution equations. Its use in optimal control provides a welcome update to Fattorini¿s work on time-optimal and norm-optimal control problems. By discussing the best way of representing various control problems and equivalence among them, this systematic study gives readers the tools they need to solve practical problems in control. After introducing preliminaries in functional analysis, evolution equations, and controllability and observability estimates, the authors present their time-optimal control framework, which consists of four elements: a controlled system, a control constraint set, a starting set, and an ending set. From there, they use their framework to address areas of recent development in time-optimal control, including the existence of admissible controls and optimal controls, Pontryagin¿s maximum principle for optimal controls, the equivalence of different optimal control problems, and bang-bang properties.This monograph will appeal to researchers and graduate students in time-optimal control theory, as well as related areas of controllability and dynamic programming. For ease of reference, the text itself is self-contained on the topic of time-optimal control. Frequent examples throughout clarify the applications of theorems and definitions, although experience with functional analysis and differential equations will be useful.

Examines a nonlinear system of parabolic partial differential equations (PDE) arising in mathematical biology and statistical mechanics. This book describes the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, and quantized blowup mechanism based on several PDE techniques.

This self-contained textbook provides the basic, abstract tools used in nonlinear analysis. The text discusses key results such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, and more.

This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations.

Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampere-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics.

This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes - non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates. Key features and topics of this self-contained, systematic exposition include:. Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions.. Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves.. Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates.. Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results. This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations - ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science.

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems.Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles.

This book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be applied to the study of parabolic problems. It presents known theorems from a novel perspective and teaches how to exploit basic techniques.