Elliptic Systems Of Phase Transition Type

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 book is an introduction to a comprehensive and unified dynamic transition theory for dissipative systems and to applications of the theory to a range of problems in the nonlinear sciences. The main objectives of this book are to introduce a general principle of dynamic transitions for dissipative systems, to establish a systematic dynamic transition theory, and to explore the physical implications of applications of the theory to a range of problems in the nonlinear sciences. The basic philosophy of the theory is to search for a complete set of transition states, and the general principle states that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. The audience for this book includes advanced graduate students and researchers in mathematics and physics as well as in other related fields. This second edition introduces a unified theory for topological phase transitions, provides a first-principle approach to statistical and quantum physics, and offers a microscopic mechanism of quantum condensates (Bose-Einstein condensation, superfluidity, and superconductivity). Reviews of first edition: “The goals of this interesting book are to derive a general principle of dynamic transitions for dissipative systems and to establish a systematic dynamic transition theory for a wide range of problems in the nonlinear sciences. ... The intended audience for this book includes students and researchers working on nonlinear problems in physics, meteorology, oceanography, biology, chemistry, and the social sciences.” (Carlo Bianca, Mathematical Reviews, December, 2014) “This is a clearly written book on numerous types of phase transitions taken in a broad sense when a dynamical dissipative system transforms from one physical state into another. ... The book is a very useful literature not only for the professionals in the field of dynamic systems and phase transitions but also for graduate students due to its interdisciplinary coverage and state-of-the-art level.” (Vladimir Čadež, zbMATH, Vol. 1285, 2014)

This IMA Volume in Mathematics and its Applications NONLINEAR EVOLUTION EQUATIONS THAT CHANGE TYPE is based on the proceedings of a workshop which was an integral part of the 1988-89 IMA program on NONLINEAR WAVES. The workshop focussed on prob lems of ill-posedness and change of type which arise in modeling flows in porous materials, viscoelastic fluids and solids and phase changes. We thank the Coordinat ing Committee: James Glimm, Daniel Joseph, Barbara Lee Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing an exciting and stimulating year-long program. We especially thank the workshop organizers, Barbara Lee Keyfitz and Michael Shearer, for their efforts in bringing together many of the major figures in those research fields in which theories for nonlinear evolution equations that change type are being developed. A vner Friedman Willard Miller, J r. ix PREFACE During the winter and spring quarters of the 1988/89 IMA Program on Non linear Waves, the issue of change of type in nonlinear partial differential equations appeared frequently. Discussion began with the January 1989 workshop on Two Phase Waves in Fluidized Beds, Sedimentation and Granular Flow; some of the papers in the proceedings of that workshop present strategies designed to avoid the appearance of change of type in models for multiphase fluid flow.

This monograph compiles, rearranges, and refines recent research results in the complex G-L theory with or without immediate applications to the theory of superconductivity. An authoritative reference for applied mathematicians, theoretical physicists and engineers interested in the quantitative description of superconductivity using Ginzburg-Landau theory.

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