Bøger i Progress in Nonlinear Differen serien

* Devoted to the motion of surfaces for which the normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process is called mean curvature flow.* Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics.

Semiconcavity is a natural generalization of concavity that retains most of the good properties known in convex analysis, but arises in a wider range of applications. This text is the first comprehensive exposition of the theory of semiconcave functions, and of the role they play in optimal control and Hamilton-Jacobi equations.The first part covers the general theory, encompassing all key results and illustrating them with significant examples. The latter part is devoted to applications concerning the Bolza problem in the calculus of variations and optimal exit time problems for nonlinear control systems. The exposition is essentially self-contained since the book includes all prerequisites from convex analysis, nonsmooth analysis, and viscosity solutions.A central role in the present work is reserved for the study of singularities. Singularities are first investigated for general semiconcave functions, then sharply estimated for solutions of Hamilton-Jacobi equations, and finally analyzed in connection with optimal trajectories of control systems.Researchers in optimal control, the calculus of variations, and partial differential equations will find this book useful as a state-of-the-art reference for semiconcave functions. Graduate students will profit from this text as it provides a handy-yet rigorous-introduction to modern dynamic programming for nonlinear control systems.

With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in superconductivity, superfluidity, and Bose--Einstein condensation. This book presents the mathematics of superconducting vortices in the framework of the acclaimed two-dimensional Ginzburg-Landau model, with or without magnetic field, and in the limit of a large Ginzburg-Landau parameter, kappa.This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in abounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.The book also provides a list of open problems and a guide to the increasingly diverse mathematical literature on Ginzburg--Landau related topics. It will benefit both pure and applied mathematicians, physicists, and graduate students having either an introductory or an advanced knowledge of the subject.

During the past decade, the mathematics of superconductivity has been the subject of intense activity. This book examines in detail the nonlinear Ginzburg–Landau functional, the model most commonly used in the study of superconductivity. Specifically covered are cases in the presence of a strong magnetic field and with a sufficiently large Ginzburg–Landau parameter kappa. Key topics and features of the work: * Provides a concrete introduction to techniques in spectral theory and partial differential equations * Offers a complete analysis of the two-dimensional Ginzburg–Landau functional with large kappa in the presence of a magnetic field * Treats the three-dimensional case thoroughly * Includes open problems Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate-level understanding of functional analysis, spectral theory, and the analysis of partial differential equations. The book also includes an overview of all nonstandard material as well as important semi-classical techniques in spectral theory that are involved in the nonlinear study of superconductivity.

This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations.Key topics include:* Operators as "sums of squares" of real and complex vector fields: both analytic hypoellipticity and regularity for very low regularity coefficients;* Nonlinear evolution equations: Navier-Stokes system, Strichartz estimates for the wave equation, instability and the Zakharov equation and eikonals;* Local solvability: its connection with subellipticity, local solvability for systems of vector fields in Gevrey classes;* Hyperbolic equations: the Cauchy problem and multiple characteristics, both positive and negative results.Graduate students at various levels as well as researchers in PDEs and related fields will find this an excellent resource.List of contributors:L. Ambrosio                            N. LernerH. Bahouri                              X. LuS. Berhanu                              J. MetcalfeJ.-M. Bony                              T. NishitaniN. Dencker                              V. PetkovS.Ervedoza                             J. RauchI. Gallagher                             M. ReissigJ. Hounie                                 L. StoyanovE. Jannelli                                D. S. TartakoffK. Kajitani                              D. TataruA. Kurganov                           F. Treves                                                G. Zampieri                                                E. Zuazua

With the goal of establishing a version for partial differential equations (PDEs) of the Aubry-Mather theory of monotone twist maps, Moser and then Bangert studied solutions of their model equations that possessed certain minimality and monotonicity properties. This monograph presents extensions of the Moser-Bangert approach that include solutions of a family of nonlinear elliptic PDEs on Rn and an Allen-Cahn PDE model of phase transitions. After recalling the relevant Moser-Bangert results, Extensions of Moser-Bangert Theory pursues the rich structure of the set of solutions of a simpler model case, expanding upon the studies of Moser and Bangert to include solutions that merely have local minimality properties. Subsequent chapters build upon the introductory results, making the monograph self contained.Part I introduces a variational approach involving a renormalized functional to characterize the basic heteroclinic solutions obtained by Bangert. Following that, Parts II and III employ these basic solutions together with constrained minimization methods to construct multitransition heteroclinic and homoclinic solutions on R×Tn-1 and R2×Tn-2, respectively, as local minima of the renormalized functional. The work is intended for mathematicians who specialize in partial differential equations and may also be used as a text for a graduate topics course in PDEs.

Since the first experimental achievement of Bose-Einstein condensates (BEC) in 1995 and the award of the Nobel Prize for Physics in 2001, the properties of these gaseous quantum fluids have been the focus of international interest in physics. This monograph is dedicated to the mathematical modelling of some specific experiments which display vortices and to a rigorous analysis of features emerging experimentally.In contrast to a classical fluid, a quantum fluid such as a Bose-Einstein condensate can rotate only through the nucleation of quantized vortices beyond some critical velocity. There are two interesting regimes: one close to the critical velocity, where there is only one vortex that has a very special shape; and another one at high rotation values, for which a dense lattice is observed. One of the key features related to superfluidity is the existence of these vortices. We address this issue mathematically and derive information on their shape, number, and location. In the dilute limit of these experiments, the condensate is well described by a mean field theory and a macroscopic wave function solving the so-called Gross-Pitaevskii equation. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. We prove existence of solutions that have properties consistent with the experimental observations. Open problems related to recent experiments are presented. The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum fluids, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.

Blowup of small data solutions for a class of quasilinear wave equations in two dimensions: an outline of the proof.- Concentration effects in critical nonlinear wave equation and scattering theory.- Lower semicontinuity of weighted path length in BV.- Time decay of Lp norms for solutions of the wave equation on exterior domains.- Sobolev embeddings in Weyl-Hörmander calculus.- About the Cauchy problem for a system of conservation laws.- Global existence of the solutions and formation of singularities for a class of hyperbolic systems.- A class of solvable operators.- On the uniqueness of the Cauchy problem under partial analyticity assumptions.- Nonlinear wave diffraction.- Caustics for dissipative semilinear oscillations.- Geometric optics and the bottom of the spectrum.- Hypoellipticity for a class of infinitely degenerate elliptic operators.- Regularity of solutions to characteristic boundary value problem for symmetric systems.