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This book is based on a course I have given five times at the
University of Michigan, beginning in 1973. The aim is to present an
introduction to a sampling of ideas, phenomena, and methods from
the subject of partial differential equations that can be presented
in one semester and requires no previous knowledge of differential
equations. The problems, with hints and discussion, form an
important and integral part of the course. In our department,
students with a variety of specialties-notably differen tial
geometry, numerical analysis, mathematical physics, complex
analysis, physics, and partial differential equations-have a need
for such a course. The goal of a one-term course forces the
omission of many topics. Everyone, including me, can find fault
with the selections that I have made. One of the things that makes
partial differential equations difficult to learn is that it uses a
wide variety of tools. In a short course, there is no time for the
leisurely development of background material. Consequently, I
suppose that the reader is trained in advanced calculus, real
analysis, the rudiments of complex analysis, and the language
offunctional analysis. Such a background is not unusual for the
students mentioned above. Students missing one of the "essentials"
can usually catch up simultaneously. A more difficult problem is
what to do about the Theory of Distributions.
This IMA Volume in Mathematics and its Applications QUASICLASSICAL
METHODS is based on the proceedings of a very successful one-week
workshop with the same title, which was an integral part of the
1994-1995 IMA program on "Waves and Scattering." We would like to
thank Jeffrey Rauch and Barry Simon for their excellent work as
organizers of the meeting. We also take this opportunity to thank
the National Science Foun dation (NSF), the Army Research Office
(ARO) and the Office of Naval Research (ONR), whose financial
support made the workshop possible. A vner Friedman Robert Gulliver
v PREFACE There are a large number of problems where qualitative
features of a partial differential equation in an appropriate
regime are determined by the behavior of an associated ordinary
differential equation. The example which gives the area its name is
the limit of quantum mechanical Hamil tonians (Schrodinger
operators) as Planck's constant h goes to zero, which is determined
by the corresponding classical mechanical system. A sec ond example
is linear wave equations with highly oscillatory initial data. The
solutions are described by geometric optics whose centerpiece are
rays which are solutions of ordinary differential equations
analogous to the clas sical mechanics equations in the example
above. Much recent work has concerned with understanding terms
beyond the leading term determined by the quasi classical limit.
Two examples of this involve Weyl asymptotics and the large-Z limit
of atomic Hamiltonians, both areas of current research."
This volume contains a multiplicity of approaches brought to bear
on problems varying from the formation of caustics and the
propagation of waves at a boundary, to the examination of viscous
boundary layers. It examines the foundations of the theory of high-
frequency electromagnetic waves in a dielectric or semiconducting
medium. Nor are unifying themes entirely absent from nonlinear
analysis: one chapter considers microlocal analysis, including
paradifferential operator calculus, on Morrey spaces, and
connections with various classes of partial differential equations.
This IMA Volume in Mathematics and its Applications MICROLOCAL
ANALYSIS AND NONLINEAR WAVES is based on the proceedings of a
workshop which was an integral part of the 1988- 1989 IMA program
on "Nonlinear Waves". We thank Michael Beals, Richard Melrose and
Jeffrey Rauch for organizing the meeting and editing this
proceedings volume. We also take this opportunity to thank the
National Science Foundation whose financial support made the
workshop possible. A vner Friedman Willard Miller, Jr. PREFACE
Microlocal analysis is natural and very successful in the study of
the propagation of linear hyperbolic waves. For example consider
the initial value problem Pu = f E e'(RHd), supp f C {t ;::: O} u =
0 for t < o. If P( t, x, Dt,x) is a strictly hyperbolic operator
or system then the singular support of f gives an upper bound for
the singular support of u (Courant-Lax, Lax, Ludwig), namely
singsupp u C the union of forward rays passing through the singular
support of f.
This IMA Volume in Mathematics and its Applications QUASICLASSICAL
METHODS is based on the proceedings of a very successful one-week
workshop with the same title, which was an integral part of the
1994-1995 IMA program on "Waves and Scattering." We would like to
thank Jeffrey Rauch and Barry Simon for their excellent work as
organizers of the meeting. We also take this opportunity to thank
the National Science Foun dation (NSF), the Army Research Office
(ARO) and the Office of Naval Research (ONR), whose financial
support made the workshop possible. A vner Friedman Robert Gulliver
v PREFACE There are a large number of problems where qualitative
features of a partial differential equation in an appropriate
regime are determined by the behavior of an associated ordinary
differential equation. The example which gives the area its name is
the limit of quantum mechanical Hamil tonians (Schrodinger
operators) as Planck's constant h goes to zero, which is determined
by the corresponding classical mechanical system. A sec ond example
is linear wave equations with highly oscillatory initial data. The
solutions are described by geometric optics whose centerpiece are
rays which are solutions of ordinary differential equations
analogous to the clas sical mechanics equations in the example
above. Much recent work has concerned with understanding terms
beyond the leading term determined by the quasi classical limit.
Two examples of this involve Weyl asymptotics and the large-Z limit
of atomic Hamiltonians, both areas of current research."
This volume contains a multiplicity of approaches brought to bear
on problems varying from the formation of caustics and the
propagation of waves at a boundary, to the examination of viscous
boundary layers. It examines the foundations of the theory of high-
frequency electromagnetic waves in a dielectric or semiconducting
medium. Nor are unifying themes entirely absent from nonlinear
analysis: one chapter considers microlocal analysis, including
paradifferential operator calculus, on Morrey spaces, and
connections with various classes of partial differential equations.
The objective of this book is to present an introduction to the ideas, phenomena, and methods of partial differential equations. This material can be presented in one semester and requires no previous knowledge of differential equations, but assumes the reader to be familiar with advanced calculus, real analysis, the rudiments of complex analysis, and thelanguage of functional analysis. Topics discussed in the text include elliptic, hyperbolic, and parabolic equations, the energy method, maximum principle, and the Fourier Transform. The text features many historical and scientific motivations and applications. Included throughout are exercises, hints, and discussions which form an important and integral part of the course.
This book introduces graduate students and researchers in
mathematics and the sciences to the multifaceted subject of the
equations of hyperbolic type, which are used, in particular, to
describe propagation of waves at finite speed. Among the topics
carefully presented in the book are nonlinear geometric optics, the
asymptotic analysis of short wavelength solutions, and nonlinear
interaction of such waves. Studied in detail are the damping of
waves, resonance, dispersive decay, and solutions to the
compressible Euler equations with dense oscillations created by
resonant interactions. Many fundamental results are presented for
the first time in a textbook format. In addition to dense
oscillations, these include the treatment of precise speed of
propagation and the existence and stability questions for the three
wave interaction equations. One of the strengths of this book is
its careful motivation of ideas and proofs, showing how they evolve
from related, simpler cases. This makes the book quite useful to
both researchers and graduate students interested in hyperbolic
partial differential equations. Numerous exercises encourage active
participation of the reader. The author is a professor of
mathematics at the University of Michigan. A recognized expert in
partial differential equations, he has made important contributions
to the transformation of three areas of hyperbolic partial
differential equations: nonlinear microlocal analysis, the control
of waves, and nonlinear geometric optics.
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