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The term "weakly differentiable functions" in the title refers to
those inte n grable functions defined on an open subset of R whose
partial derivatives in the sense of distributions are either LP
functions or (signed) measures with finite total variation. The
former class of functions comprises what is now known as Sobolev
spaces, though its origin, traceable to the early 1900s, predates
the contributions by Sobolev. Both classes of functions, Sobolev
spaces and the space of functions of bounded variation (BV func
tions), have undergone considerable development during the past 20
years. From this development a rather complete theory has emerged
and thus has provided the main impetus for the writing of this
book. Since these classes of functions play a significant role in
many fields, such as approximation theory, calculus of variations,
partial differential equations, and non-linear potential theory, it
is hoped that this monograph will be of assistance to a wide range
of graduate students and researchers in these and perhaps other
related areas. Some of the material in Chapters 1-4 has been
presented in a graduate course at Indiana University during the
1987-88 academic year, and I am indebted to the students and
colleagues in attendance for their helpful comments and
suggestions."
The major thrust of this book is the analysis of pointwise behavior
of Sobolev functions of integer order and BV functions (functions
whose partial derivatives are measures with finite total
variation). The development of Sobolev functions includes an
analysis of their continuity properties in terms of Lebesgue
points, approximate continuity, and fine continuity as well as a
discussion of their higher order regularity properties in terms of
Lp-derivatives. This provides the foundation for further results
such as a strong approximation theorem and the comparison of Lp and
distributional derivatives. Also included is a treatment of
Sobolev-PoincarA(c) type inequalities which unifies virtually all
inequalities of this type. Although the techniques required for the
discussion of BV functions are completely different from those
required for Sobolev functions, there are similarities between
their developments such as a unifying treatment of PoincarA(c)-type
inequalities for BV functions. This book is intended for graduate
students and researchers whose interests may include aspects of
approximation theory, the calculus of variations, partial
differential equations, potential theory and related areas. The
only prerequisite is a standard graduate course in real analysis
since almost all of the material is accessible through real
variable techniques.
The 39 papers in this collection are devoted mostly to the exact
mathematical analysis of problems in continuum mechanics, but also
to problems of a purely mathematical nature mainly connected to
partial differential equations from continuum physics. All the
papers are dedicated to J. Serrin and were originally published in
the "Archive of Rational Mechanics and Analysis."
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