Series: Birkhauser Advanced Texts / Basler Lehrbucher
2007, Approx. 160 p., 5 illus., Hardcover
ISBN-10: 0-8176-4535-7
ISBN-13: 978-0-8176-4535-9
About this textbook
This textbook introduces variational methods and their
applications to differential equations to graduate students and
researchers interested in differential equations and nonlinear
analysis.
The topics covered include: Minimizations; Deformations results;
The mountain-pass and saddle-point theorems; Critical points
under constraints; Issues of compactness.
The straightforward and systematic presentation includes many
exercises and examples to motivate the study of variational
methods.
Table of contents
Preface.- Introduction.- Critical Points Via Minimization.- The
Deformation Theorem.- The Mountain-Pass Theorem.- The Saddle-Point
Theorem.- Critical Points under Constraints.- A Duality Principle.-
Critical Points under Symmetries.- Problems with an S1-Symmetry.-
Problems with Lack of Compactness.- Lack of Compactness for
Bounded \Omega.-Index
Series: Progress in Mathematics , Vol. 261
2007, Approx. 445 p., Hardcover
ISBN-10: 0-8176-4537-3
ISBN-13: 978-0-8176-4537-3
About this book
This comprehensive monograph is devoted to the study of
polynomially convex sets, which play a key role in the theory of
functions of several complex variables
Important features of Polynomial Convexity: Presents the general
properties of polynomially convex sets with particular attention
to the theory of the hulls of one-dimensional sets; Motivates the
theory with numerous examples and counterexamples, which serve to
illustrate the general theory and to delineate its boundaries;
Examines in considerable detail questions of uniform
approximation, especially on totally real sets, for the most part
on compact sets but with some attention to questions of global
approximation on noncompact sets; Discusses important
applications, e.g., to the study of analytic varieties and to the
theory of removable singularities for CR functions; Requires of
the reader a solid background in real and complex analysis
together with some previous experience with the theory of
functions of several complex variables as well as the elements of
functional analysis.
This beautiful exposition of a rich and complex theory, which
contains much material not available in other texts and which is
destined to be the standard reference for many years, will appeal
to all those with an interest in multivariate complex analysis.
Table of contents
Preface.
Introduction. Polynomial convexity. Uniform algebras.
Plurisubharmonic fuctions. The Cauchy-Fantappie Integral. The
Oka?Weil Theorem. Some examples. Hulls with no analytic structure.-
Some General Properties of Polynomially Convex Sets. Applications
of the Cousin problems. Two characterizations of polynomially
convex sets. Applications of Morse theory and algebraic topology.
Convexity in Stein manifolds.-
Sets of Finite Length. Introduction. One-dimensional varieties.
Geometric preliminaries. Function-theoretic preliminaries.
Subharmonicity results. Analytic structure in hulls. Finite area.
The continuation of varieties.-
Sets of Class A1. Introductory remarks. Measure-theoretic
preliminaries. Sets of class A1. Finite area. Stokesfs Theorem.
The multiplicity function. Counting the branches.-
Further Results. Isoperimetry. Removable singularities. Surfaces
in strictly pseudoconvex boundaries.-
Approximation. Totally real manifolds. Holomorphically convex
sets. Approximation on totally real manifolds. Some tools from
rational approximation. Algebras on surfaces. Tangential
approximation.-
Varieties in Strictly Pseudoconvex Domains. Interpolation.
Boundary regularity. Uniqueness.-
Examples and Counter Examples. Unions of planes and balls.
Pluripolar graphs. Deformations. Sets with symmetry.-
Bibliography. Index.
Series: Birkhauser Advanced Texts / Basler Lehrbucher
2008, Approx. 400 p., 20 illus., Hardcover
ISBN-10: 0-8176-4139-4
ISBN-13: 978-0-8176-4139-9
About this book
This highly original work, written by the creators of the
multivariable theory of automorphisms, is a rich tapestry of
themes and concepts, and a comprehensive treatment of an
important area of mathematics. From Poincare's work on
biholomorphic inequivalence in 1906, it became clear that the
structures of the automorphism groups of domains in multi-dimensional
complex space are more complex, and more interesting, than those
in the complex plane. The authors build on this theme and trace
the evolution of the classical theory to the modern theory, which
is today a cornerstone of geometric analysis.
The text begins with an introductory chapter on the concept of an
automorphism group in which the theory in one complex variable is
presented, emphasizing the classical ideas of Schwarz, Jobe, and
others. Also examined is the theory of planar domains of multiple
but finite connectivity, principally develped by Heins in the
1940s and 1950s. The authors treatment progresses to the theory
in several complex variables with the so-called "classical
domains" of E. Cartan, the Siegel domains of type I, II, and
III, and the more modern theory of automorphism groups of
smoothly bounded domains.
Table of contents
Preface * The Concept of an Automorphism Group * Complexs
Dimension One * General Priciples of Automorphism Groups *
Domains in Complex Space with Smooth Boundary * Scaling *
Examples * Reinhardt Domains * Bibliography * Index
Series: Cornerstones
2007, Approx. 335 p., 10 illus., Hardcover
ISBN-10: 0-8176-4551-9
ISBN-13: 978-0-8176-4551-9
About this textbook
This self-contained text offers an elementary introduction to
partial differential equations (pdes), primarily focusing on
linear equations, but also providing some perspective on
nonlinear equations. The classical treatment is mathematically
rigorous with a generally theoretical layout, though indications
to some of the physical origins of pdes are made throughout in
references to potential theory, similarity solutions for the
porous medium equation, generalized Riemann problems, and others.
The material begins with a focus on the Cauchy?Kowalewski
theorem, discussing the notion of characteristic surfaces to
classify pdes. Next, the Laplace equation and connected elliptic
theory are treated, as well as integral equations and solutions
to eigenvalue problems. The heat equation and related parabolic
theory are then presented, followed by the wave equation in its
basic aspects. An introduction to conservation laws, the
uniqueness theorem, viscosity solutions, ill-posed problems, and
nonlinear equations of first order round out the key subject
matter.
Large parts of this revised second edition have been streamlined
and rewritten to incorporate years of classroom feedback, correct
errors, and improve clarity. Most of the necessary background
material has been incorporated into the complements and certain
nonessential topics have been given reduced attention (noticeably,
numerical methods) to improve the flow of presentation.
The exposition is replete with examples, problems and solutions
that compliment the material to enhance understanding and
solidify comprehension. The only prerequisites are advanced
differential calculus and some basic Lp theory. The work can
serve as a text for advanced undergraduates and graduate students
in mathematics, physics, engineering, and the natural sciences,
as well as an excellent reference for applied mathematicians and
mathematical physicists.
Table of contents
Preface to the Second Edition.- Preface to the First Edition.-
Preliminaries.- Quasi-Linear Equations.- The LaPlace Equation.-
Boundary Value Problems by Double Layer Potentials.- Integral
Equations and Eigenvalue Problems.- The Heat Equation.- The Wave
Equation.- Quasi-Linear Equations of First Order.- Non-Linear
Equations of First Order.- References.- Index.