This volume is a collection of articles presented at the Workshop for Nonlinear Analysis held in Joao Pessoa, Brazil, in September 2012. The influence of Bernhard Ruf, to whom this volume is dedicated on the occasion of his 60th birthday, is perceptible throughout the collection by the choice of themes and techniques. The many contributors consider modern topics in the calculus of variations, topological methods and regularity analysis, together with novel applications of partial differential equations. In keeping with the tradition of the workshop, emphasis is given to elliptic operators inserted in different contexts, both theoretical and applied. Topics include semi-linear and fully nonlinear equations and systems with different nonlinearities, at sub and supercritical exponents, with spectral interactions of Ambrosetti-Prodi type. Also treated are analytic aspects as well as applications such as diffusion problems in mathematical genetics and finance and evolution equations related to electromechanical devices."

This volume contains lectures and invited papers from the Focus Program on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" held at the Fields Institute from July 31-August 18, 2017. The conference brought together researchers in completely integrable systems and PDE with the goal of advancing the understanding of qualitative and long-time behavior in dispersive nonlinear equations. The program included Percy Deift's Coxeter lectures, which appear in this volume together with tutorial lectures given during the first week of the focus program. The research papers collected here include new results on the focusing nonlinear Schroedinger (NLS) equation, the massive Thirring model, and the Benjamin-Bona-Mahoney equation as dispersive PDE in one space dimension, as well as the Kadomtsev-Petviashvili II equation, the Zakharov-Kuznetsov equation, and the Gross-Pitaevskii equation as dispersive PDE in two space dimensions. The Focus Program coincided with the fiftieth anniversary of the discovery by Gardner, Greene, Kruskal and Miura that the Korteweg-de Vries (KdV) equation could be integrated by exploiting a remarkable connection between KdV and the spectral theory of Schrodinger's equation in one space dimension. This led to the discovery of a number of completely integrable models of dispersive wave propagation, including the cubic NLS equation, and the derivative NLS equation in one space dimension and the Davey-Stewartson, Kadomtsev-Petviashvili and Novikov-Veselov equations in two space dimensions. These models have been extensively studied and, in some cases, the inverse scattering theory has been put on rigorous footing. It has been used as a powerful analytical tool to study global well-posedness and elucidate asymptotic behavior of the solutions, including dispersion, soliton resolution, and semiclassical limits.

The conjugate gradient method is a powerful tool for the iterative solution of self-adjoint operator equations in Hilbert space.This volume summarizes and extends the developments of the past decade concerning the applicability of the conjugate gradient method (and some of its variants) to ill posed problems and their regularization. Such problems occur in applications from almost all natural and technical sciences, including astronomical and geophysical imaging, signal analysis, computerized tomography, inverse heat transfer problems, and many more This Research Note presents a unifying analysis of an entire family of conjugate gradient type methods. Most of the results are as yet unpublished, or obscured in the Russian literature. Beginning with the original results by Nemirovskii and others for minimal residual type methods, equally sharp convergence results are then derived with a different technique for the classical Hestenes-Stiefel algorithm. In the final chapter some of these results are extended to selfadjoint indefinite operator equations. The main tool for the analysis is the connection of conjugate gradient type methods to real orthogonal polynomials, and elementary properties of these polynomials. These prerequisites are provided in a first chapter. Applications to image reconstruction and inverse heat transfer problems are pointed out, and exemplarily numerical results are shown for these applications.

This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

During last decade significant progress has been made in the oil indus try by using soft computing technology. Underlying this evolving technology there have, been ideas transforming the very language we use to describe problems with imprecision, uncertainty and partial truth. These developments offer exciting opportunities, but at the same time it is becoming clearer that further advancements are confronted by funda mental problems. The whole idea of how human process information lies at the core of the challenge. There are already new ways of thinking about the problems within theory of perception-based information. This theory aims to understand and harness the laws of human perceptions to dramatically im prove the processing of information. A matured theory of perception-based information is likely to be proper positioned to contribute to the solution of the problems and provide all the ingredients for a revolution in science, technology and business. In this context, Berkeley Initiative in Soft Computing (BISC), Univer sity of California, Berkeley from one side and Chevron-Texaco from another formed a Technical Committee to organize a Meeting entitled "State of the Art Assessment and New Directions for Research" to understand the signifi cance of the fields accomplishments, new developments and future directions. The Technical Committee selected and invited 15 scientists (and oil indus try experts as technical committee members) from the related disciplines to participate in the Meeting, which took place at the University of California, Berkeley, and March 15-17, 2002."

The study of dynamic equations on a measure chain (time scale) goes back to its founder S. Hilger (1988), and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics. Further, the study of measure chain theory has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, and epidemic models. Key features of the book: * Introduction to measure chain theory; discussion of its usefulness in allowing for the simultaneous development of differential equations and difference equations without having to repeat analogous proofs * Many classical formulas or procedures for differential and difference equations cast in a new light * New analogues of many of the "special functions" studied * Examination of the properties of the "exponential function" on time scales, which can be defined and investigated using a particularly simple linear equation * Additional topics covered: self-adjoint equations, linear systems, higher order equations, dynamic inequalities, and symplectic dynamic systems * Clear, motivated exposition, beginning with preliminaries and progressing to more sophisticated text * Ample examples and exercises throughout the book * Solutions to selected problems Requiring only a first semester of calculus and linear algebra, Dynamic Equations on Time Scales may be considered as an interesting approach to differential equations via exposure to continuous and discrete analysis. This approach provides an early encounter with many applications in such areas as biology, physics, and engineering. Parts of the book may be used in a special topics seminar at the senior undergraduate or beginning graduate levels. Finally, the work may serve as a reference to stimulate the development of new kinds of equations with potentially new applications.

Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn damental questions concern the existence, uniqueness and regularity of so lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.

The work of Hans Lewy (1904--1988) has had a profound influence in the direction of applied mathematics and partial differential equations, in particular, from the late 1920s. Two of the particulars are well known. The Courant--Friedrichs--Lewy condition (1928), or CFL condition, was devised to obtain existence and approximation results. This condition, relating the time and spatial discretizations for finite difference schemes, is now universally employed in the simulation of solutions of equations describing propagation phenomena. Lewy's example of a linear equation with no solution (1957), with its attendant consequence that most equations have no solution, was not merely an unexpected fact, but changed the viewpoint of the entire field. Lewy made pivotal contributions in many other areas, for example, the regularity theory of elliptic equations and systems, the Monge--AmpA]re Equation, the Minkowski Problem, the asymptotic analysis of boundary value problems, and several complex variables. He was among the first to study variational inequalities. In much of his work, his underlying philosophy was that simple tools of function theory could help one understand the essential concepts embedded in an issue, although at a cost in generality. This approach was extremely successful. In this two-volume work, most all of Lewy's papers are presented, in chronological order. They are preceded by several short essays about Lewy himself, prepared by Helen Lewy, Constance Reid, and David Kinderlehrer, and commentaries on his work by Erhard Heinz, Peter Lax, Jean Leray, Richard MacCamy, FranAois Treves, and Louis Nirenberg. Additionally, there are Lewy's own remarks on the occasion of his honorarydegree from the University of Bonn.

The present monograph is devoted to the theory of the solvability in generalized functions of general boundary value problems of mathematieal physies. It is the eontinuation of the author's book [Rl], where elliptic boundary value problems have been studied in eomplete seales of spaees of generalized funetions. From the early sixties, in the works of Lions and Magenes [LiM] and Yu. Berezanskii, S. Krein and Va. Roitberg [BKR] the theorems on eomplete eolleetion of isomorphisms have been established. These theorems, roughly speaking, mean that the operator generated by an elliptie boundary value problem establishes an isomorphism between spaees of functions which 'have s derivatives' and spaees functions whieh 'have s - r derivatives' (here s is an arbitrary real number, r is the order of the elliptic problem). The dose results were also obtained by Seheehter [Sehe]. These results and some of their applieations are eontained in the book of Lions and Magenes [LiM2] (see also the survey of Magenes [MagD and Yu. Berezanskii [Ber]. Further progress in the theory under eonsideration was eonneeted, first, with the completion of the dass of elliptie problems for whieh the theorems on eomplete eolleetion of isomorphisms hold, and, henee, with the development of new methods of proving of these theorems, and, second, with the inerease of a number of applieations of the isomorphism theorems. In the author's monograph [RI] the last years' investigations on the isomorphism theorems and some of their applieations have been presented.

If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species." These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

We have considered writing the present book for a long time, since the lack of a sufficiently complete textbook about complex analysis in infinite dimensional spaces was apparent. There are, however, some separate topics on this subject covered in the mathematical literature. For instance, the elementary theory of holomorphic vector- functions.and mappings on Banach spaces is presented in the monographs of E. Hille and R. Phillips [1] and L. Schwartz [1], whereas some results on Banach algebras of holomorphic functions and holomorphic operator-functions are discussed in the books of W. Rudin [1] and T. Kato [1]. Apparently, the need to study holomorphic mappings in infinite dimensional spaces arose for the first time in connection with the development of nonlinear anal- ysis. A systematic study of integral equations with an analytic nonlinear part was started at the end ofthe 19th and the beginning ofthe 20th centuries by A. Liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the undetermined coefficients and the majorant power series methods. The most complete presentation of these methods comes from N. Nazarov. In the forties and fifties the interest in Liapunov's and Schmidt's analytic methods diminished temporarily due to the appearence of variational calculus meth- ods (M. Golomb, A. Hammerstein and others) and also to the rapid development of the mapping degree theory (J. Leray, J. Schauder, G. Birkhoff, O. Kellog and others).

Significant interest in the investigation of systems with discontinuous trajectories is explained by the development of equipment in which significant role is played by impulsive control systems and impulsive computing systems. Impulsive systems are also encountered in numerous problems of natural sciences described by mathematical models with conditions reflecting the impulsive action of external forces with pulses whose duration can be neglected. Differential equations with set-valued right-hand side arise in the investigation of evolution processes in the case of measurement errors, inaccuracy or incompleteness of information, action of bounded perturbations, violation of unique solvability conditions, etc. Differential inclusions also allow one to describe the dynamics of controlled processes and are widely used in the theory of optimal control. This monograph is devoted to the investigation of impulsive differential equations with set-valued and discontinuous right-hand sides. It is intended for researchers, lecturers, postgraduate students, and students of higher schools specialized in the field of the theory of differential equations, the theory of optimal control, and their applications.

This book explains mathematical theories of a collection of stochastic partial differential equations and their dynamical behaviors. Based on probability and stochastic process, the authors discuss stochastic integrals, Ito formula and Ornstein-Uhlenbeck processes, and introduce theoretical framework for random attractors. With rigorous mathematical deduction, the book is an essential reference to mathematicians and physicists in nonlinear science. Contents: Preliminaries The stochastic integral and Ito formula OU processes and SDEs Random attractors Applications Bibliography Index

The present monograph defines, interprets and uses the matrix of partial derivatives of the state vector with applications for the study of some common categories of engineering. The book covers broad categories of processes that are formed by systems of partial derivative equations (PDEs), including systems of ordinary differential equations (ODEs). The work includes numerous applications specific to Systems Theory based on Mpdx, such as parallel, serial as well as feed-back connections for the processes defined by PDEs. For similar, more complex processes based on Mpdx with PDEs and ODEs as components, we have developed control schemes with PID effects for the propagation phenomena, in continuous media (spaces) or discontinuous ones (chemistry, power system, thermo-energetic) or in electro-mechanics (railway - traction) and so on. The monograph has a purely engineering focus and is intended for a target audience working in extremely diverse fields of application (propagation phenomena, diffusion, hydrodynamics, electromechanics) in which the use of PDEs and ODEs is justified.

This monograph is devoted to the systematic presentation of the method of singular quadratic forms in the perturbation theory of self-adjoint operators. The concept of a singular (nowhere closable) quadratic form, a key notion of the present volume, is treated from different points of view such as definition, properties, relations with regular (closable) quadratic forms, operator representation, classification in the scale of Hilbert spaces and especially as an object carrying a singular perturbation for Hamiltonians. The main idea is to interpret singular quadratic form in the role of an abstract boundary condition for self-adjoint extension. Various aspects of the singularity principle are investigated, such as the construction of singularly perturbed operators, higher powers of perturbed operators, the transition to a new orthogonally extended state space, as well as approximation and regularization. Furthermore, applications dealing with singular Wick monomials in the Fock space and mathematical scattering theory are included. Audience: This book will be of interest to students and researchers whose work involves functional analysis, operator theory and quantum field theory.

This volume contains the proceedings of the 22nd International Conference on Difference Equations and Applications, held at Osaka Prefecture University, Osaka, Japan, in July 2016. The conference brought together both experts and novices in the theory and applications of difference equations and discrete dynamical systems. The volume features papers in difference equations and discrete dynamical systems with applications to mathematical sciences and, in particular, mathematical biology and economics. This book will appeal to researchers, scientists, and educators who work in the fields of difference equations, discrete dynamical systems, and their applications.

The construction of solutions of singularly perturbed systems of equations and boundary value problems that are characteristic for the mechanics of thin-walled structures are the main focus of the book. The theoretical results are supplemented by the analysis of problems and exercises. Some of the topics are rarely discussed in the textbooks, for example, the Newton polyhedron, which is a generalization of the Newton polygon for equations with two or more parameters. After introducing the important concept of the index of variation for functions special attention is devoted to eigenvalue problems containing a small parameter. The main part of the book deals with methods of asymptotic solutions of linear singularly perturbed boundary and boundary value problems without or with turning points, respectively. As examples, one-dimensional equilibrium, dynamics and stability problems for rigid bodies and solids are presented in detail. Numerous exercises and examples as well as vast references to the relevant Russian literature not well known for an English speaking reader makes this a indispensable textbook on the topic.

In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation, as well as for the existence and persistence of fibrations of these invariant manifolds. Their techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Fenichels results. As such, they may be applied to a broad class of infinite dimensional dynamical systems.

This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element and finite volume methods, interweaving theory and applications throughout. Extensive exercises are provided throughout the text. Graduate students in mathematics, engineering and physics will find this book useful.

This book deals with linear functional differential equations and operator theory methods for their investigation. The main topics are: the equivalence of the input-output stability of the equation "L"x = &mathsf; and the invertibility of the operator "L" in the class of casual operators; the equivalence of input-output and exponential stability; the equivalence of the dichotomy of solutions for the homogeneous equation "L"x = 0 and the invertibility of the operator "L"; the properties of Green's function; the independence of the stability of an equation from the norm on the space of solutions; shift invariant functional differential equations in Banach space; the possibility of the reduction of an equation of neutral type to an equation of retarded type; special full subalgebras of integral and difference operators, and operators with unbounded memory; and the analogue of Fredholm's alternative for operators with almost periodic coefficients where one-sided invertibility implies two-sided invertibility. Audience: This monograph will be of interest to students and researchers working in functional differential equations and operator theory and is recommended for graduate level courses.

This book gathers a collection of refereed articles containing original results reporting the recent contributions of the lectures and communications presented at the Free Boundary Problems Conference that took place at the University of Coimbra, Portugal, from June 7 to 12, 2005 (FBP2005). They deal with the mathematics of a broad class of models and problems involving nonlinear partial differential equations arising in physics, engineering, biology and finance. Among the main topics, the talks considered free boundary problems in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in image processing, in financial mathematics or in computations for inter-scale problems.

The mathematical analysis and fine properties of solutions and interfaces in free boundary problems have been an active subject in the last three decades and their mathematical understanding continues to be an important interdisciplinary

tool for the scientific applications, on one hand, and an intrinsic aspect of the current development of several important mathematical disciplines.

This volume contains surveys as well as research articles broadly centered on spectral analysis. Topics range from spectral continuity for magnetic and pseudodifferential operators to localization in random media, from the stability of matter to properties of Aharonov-Bohm and Quantum Hall Hamiltonians, from waveguides and resonances to supersymmetric models and dissipative fermion systems. This is the first of a series of volumes reporting every two years on recent progress in spectral theory. "

The interest in inverse problems of spectral analysis has increased considerably in recent years due to the applications to important non-linear equations in mathematical physics. This monograph is devoted to the detailed theory of inverse problems and methods of their solution for the Sturm-Liouville case. Chapters 1--6 contain proofs which are, in many cases, very different from those known earlier. Chapters 4--6 are devoted to inverse problems of quantum scattering theory with attention being focused on physical applications. Chapters 7--11 are based on the author's recent research on the theory of finite- and infinite-zone potentials. A chapter discussing the applications to the Korteweg--de Vries problem is also included. This monograph is important reading for all researchers in the field of mathematics and physics.

This book deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional dynamical systems methods for elliptic problems in unbounded domains as well as finite dimensional reduction of their dynamics requires new ideas and tools. To this end, both a trajectory dynamical systems approach and new Liouville type results for the solutions of some class of elliptic equations are used. The work also uses symmetry and monotonicity results for nonnegative solutions in order to characterize an asymptotic profile of solutions and compares a pure elliptic partial differential equations approach and a dynamical systems approach. The new results obtained will be particularly useful for mathematical biologists.

The position taken in this collection of pedagogically written essays is that conjugate gradient algorithms and finite element methods complement each other extremely well.

Via their combinations practitioners have been able to solve complicated, direct and inverse, multidemensional problems modeled by ordinary or partial differential equations and inequalities, not necessarily linear, optimal control and optimal design being part of these problems.

The aim of this book is to present both methods in the context of complicated problems modeled by linear and nonlinear partial differential equations, to provide an in-depth discussion on their implementation aspects. The authors show that conjugate gradient methods and finite element methods apply to the solution of real-life problems. They address graduate students as well as experts in scientific computing.