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PMThis is the first of two books on methods and techniques in the
calculus of variations. Contemporary arguments are used throughout
the text to streamline and present in a unified way classical
results, and to provide novel contributions at the forefront of the
theory. This book addresses fundamental questions related to lower
semicontinuity and relaxation of functionals within the
unconstrained setting, mainly in L p spaces. It prepares the ground
for the second volume where the variational treatment of
functionals involving fields and their derivatives will be
undertaken within the framework of Sobolev spaces. This book is
self-contained. All the statements are fully justified and proved,
with the exception of basic results in measure theory, which may be
found in any good textbook on the subject. It also contains several
exercises. Therefore,it may be used both as a graduate textbook as
well as a reference text for researchers in the field. Irene
Fonseca is the Mellon College of Science Professor of Mathematics
and is currently the Director of the Center for Nonlinear Analysis
in the Department of Mathematical Sciences at Carnegie Mellon
University. Her research interests lie in the areas of continuum
mechanics, calculus of variations, geometric measure theory and
partial differential equations. Giovanni Leoni is also a professor
in the Department of Mathematical Sciences at Carnegie Mellon
University. He focuses his research on calculus of variations,
partial differential equations and geometric measure theory with
special emphasis on applications to problems in continuum mechanics
and in materials scienc
This is the first of two books on methods and techniques in the
calculus of variations. Contemporary arguments are used throughout
the text to streamline and present in a unified way classical
results, and to provide novel contributions at the forefront of the
theory. This book addresses fundamental questions related to lower
semicontinuity and relaxation of functionals within the
unconstrained setting, mainly in L^p spaces. It prepares the ground
for the second volume where the variational treatment of
functionals involving fields and their derivatives will be
undertaken within the framework of Sobolev spaces. This book is
self-contained. All the statements are fully justified and proved,
with the exception of basic results in measure theory, which may be
found in any good textbook on the subject. It also contains several
exercises. Therefore,it may be used both as a graduate textbook as
well as a reference text for researchers in the field. Irene
Fonseca is the Mellon College of Science Professor of Mathematics
and is currently the Director of the Center for Nonlinear Analysis
in the Department of Mathematical Sciences at Carnegie Mellon
University. Her research interests lie in the areas of continuum
mechanics, calculus of variations, geometric measure theory and
partial differential equations. Giovanni Leoni is also a professor
in the Department of Mathematical Sciences at Carnegie Mellon
University. He focuses his research on calculus of variations,
partial differential equations and geometric measure theory with
special emphasis on applications to problems in continuum mechanics
and in materials science.
In this book we study the degree theory and some of its
applications in analysis. It focuses on the recent developments of
this theory for Sobolev functions, which distinguishes this book
from the currently available literature. We begin with a thorough
study of topological degree for continuous functions. The contents
of the book include: degree theory for continuous functions, the
multiplication theorem, Hopf`s theorem, Brower`s fixed point
theorem, odd mappings, Jordan`s separation theorem. Following a
brief review of measure theory and Sobolev functions and study
local invertibility of Sobolev functions. These results are put to
use in the study variational principles in nonlinear elasticity.
The Leray-Schauder degree in infinite dimensional spaces is
exploited to obtain fixed point theorems. We end the book by
illustrating several applications of the degree in the theories of
ordinary differential equations and partial differential equations.
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