This textbook on the calculus of variations leads the reader from
the basics to modern aspects of the theory. One-dimensional
problems and the classical issues such as Euler-Lagrange equations
are treated, as are Noether's theorem, Hamilton-Jacobi theory, and
in particular geodesic lines, thereby developing some important
geometric and topological aspects. The basic ideas of optimal
control theory are also given. The second part of the book deals
with multiple integrals. After a review of Lebesgue integration,
Banach and Hilbert space theory and Sobolev spaces (with complete
and detailed proofs), there is a treatment of the direct methods
and the fundamental lower semicontinuity theorems. Subsequent
chapters introduce the basic concepts of the modern calculus of
variations, namely relaxation, Gamma convergence, bifurcation
theory and minimax methods based on the Palais-Smale condition. The
prerequisites are knowledge of the basic results from calculus of
one and several variables. After having studied this book, the
reader will be well equipped to read research papers in the
calculus of variations.
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