Presents the state of the art in PDEs, including the latest research and short courses accessible to graduate students.

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis

The unique feature of this book is that it considers the theory of partial differential equations in mathematical physics as the language of continuous processes, that is, as an interdisciplinary science that treats the hierarchy of mathematical phenomena as reflections of their physical counterparts. Special attention is drawn to tracing the development of these mathematical phenomena in different natural sciences, with examples drawn from continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics. At the same time, the authors trace the interrelation between the different types of problems - elliptic, parabolic, and hyperbolic - as the mathematical counterparts of stationary and evolutionary processes. This combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs, one that will be appreciated by both students and researchers alike.

The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations.The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

In this edited volume leaders in the field of partial differential equations present recent work on topics in PDEs arising from geometry and physics. The papers originate from a 2015 research school organized by CIMPA and MIMS in Hammamet, Tunisia to celebrate the 60th birthday of the late Professor Abbas Bahri. The opening chapter commemorates his life and work. While the research presented in this book is cutting-edge, the treatment throughout is at a level accessible to graduate students. It includes short courses offering readers a unique opportunity to learn the state of the art in evolution equations and mathematical models in physics, which will serve as an introduction for students and a useful reference for established researchers. Finally, the volume includes many open problems to inspire the next generation.

This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS BY H. BAT EM AN, M. A., PH. D. Late Fellow of Trinity College, Cambridge Professor of Mathematics, Theoretical Physics and Aeronautics, California Institute of Technology, Pasadena, California NEW YORK DOVER PUBLICATIONS 1944 First Edition 1932 First American Edition 1944 By special arrangement with the Cambridge University Press and The Macmillan Co. Printed in the U. S. A. Dedicated to MY MOTHER CONTENTS PREFACE page xiii INTRODUCTION xv-xxii CHAPTER I THE CLASSICAL EQUATIONS 1-11-1-14. Uniform motion, boundary conditions, problems, a passage to the limit. 1-7 1-15-1-19. Fouriers theorem, Fourier constants, Cesaros method of summation, Parsevals theorem, Fourier series, the expansion of the integral of a bounded function which is continuous bit by bit. . 7-16 1-21-1-25. The bending of a beam, the Greens function, the equation of three moments, stability of a strut, end conditions, examples. 16-25 1 31-1-36. F ee undamped vibrations, simple periodic motion, simultaneous linear equations, the Lagrangian equations of motion, normal vibrations, com pound pendulum, quadratic forms, Hermit ian forms, examples. 25-40 1-41-1 - 42. Forced oscillations, residual oscillation, examples. 40-44 1-43. Motion with a resistance proportional to the velocity, reduction to alge braic equations. 44 d7 1-44. The equation of damped vibrations, instrumental records. 47-52 1-45-1 - 46. The dissipation function, reciprocal relations. 52-54 1-47-1-49. Fundamental equations of electric circuit theory, Cauchys method of solving a linear equation, Heavisides expansion. 54-6Q 1-51 1-56. The simple wave-equation, wave propagation, associated equations, transmission of vibrations, vibration of a building, vibration of a string, torsional oscillations of a rod, plane waves of sound, waves in a canal, examples. 60-73 1-61-1 - 63. Conjugate functions and systems of partial differential equations, the telegraphic equation, partial difference equations, simultaneous equations involving high derivatives, examplu. 73-77 1-71-1-72. Potentials and stream-functions, motion of a fluid, sources and vortices, two-dimensional stresses, geometrical properties of equipotentials and lines of force, method of inversion, examples. 77-90 1-81-1-82. The classical partial differential equations for Euclidean space, Laplaces equation, systems of partial differential equations of the first order fchich lead to the classical equations, elastic equilibrium, equations leading to the uations of wave-motion, 90-95 S 1 91. Primary solutions, Jacobis theorem, examples. 95-100 1 92. The partial differential equation of the characteristics, bicharacteristics and rays. 101-105 1 93-1 94. Primary solutions of the second grade, primitive solutions of the wave-equation, primitive solutions of Laplaces equation. 105-111 1-95. Fundamental solutions, examples. 111-114 viii Contents CHAPTER n APPLICATIONS OF THE INTEGRAL THEOREMS OF GREEN AND STOKES 2 11-2-12. Greens theorem, Stokes s theorem, curl of a vector, velocity potentials, equation of continuity. pages 116-118 2-13-2-16. The equation of the conduction of heat, diffusion, the drying of wood, the heating of a porous body by a warm fluid, Laplaces method, example. 118-125 2-21-2 22. Riemanns method, modified equation of diffusion, Greens func tions, examples. 126-131 f 2-23-2 26. Green s theorem for a general lineardifferential equation of the second order, characteristics, classification of partial differential equations of the second order, a property of equations of elliptic type, maxima and minima of solutions. 131-138 2-31-2-32. Greens theorem for Laplaces equation, Greens functions, reciprocal relations. 138-144 2-33-2-34. Partial difference equations, associated quadratic form, the limiting process, inequalities, properties of the limit function. 144-152 2-41-2-42...

An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis. Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs. Provides an accessible yet rigorous introduction to partial differential equations Draws connections to advanced topics in analysis Covers applications to continuum mechanics An electronic solutions manual is available only to professors An online illustration package is available to professors