This book presents a systematic approach to a solution theory for
linear partial differential equations developed in a Hilbert space
setting based on a Sobolev lattice structure, a simple extension of
the well-established notion of a chain (or scale) of Hilbert
spaces. The focus on a Hilbert space setting (rather than on an
apparently more general Banach space) is not a severe constraint,
but rather a highly adaptable and suitable approach providing a
more transparent framework for presenting the main issues in the
development of a solution theory for partial differential
equations. In contrast to other texts on partial differential
equations, which consider either specific equation types or apply a
collection of tools for solving a variety of equations, this book
takes a more global point of view by focusing on the issues
involved in determining the appropriate functional analytic setting
in which a solution theory can be naturally developed. Applications
to many areas of mathematical physics are also presented. The book
aims to be largely self-contained. Full proofs to all but the most
straightforward results are provided, keeping to a minimum
references to other literature for essential material. It is
therefore highly suitable as a resource for graduate courses and
also for researchers, who will find new results for particular
evolutionary systems from mathematical physics.
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