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Parametrized Measures and Variational Principles (Paperback, Softcover reprint of the original 1st ed. 1997)
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Parametrized Measures and Variational Principles (Paperback, Softcover reprint of the original 1st ed. 1997)
Series: Progress in Nonlinear Differential Equations and Their Applications, 30
Expected to ship within 10 - 15 working days
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Weak convergence is a basic tool of modern nonlinear analysis
because it enjoys the same compactness properties that finite
dimensional spaces do: basically, bounded sequences are weak
relatively compact sets. Nonetheless, weak conver gence does not
behave as one would desire with respect to nonlinear functionals
and operations. This difficulty is what makes nonlinear analysis
much harder than would normally be expected. Parametrized measures
is a device to under stand weak convergence and its behavior with
respect to nonlinear functionals. Under suitable hypotheses, it
yields a way of representing through integrals weak limits of
compositions with nonlinear functions. It is particularly helpful
in comprehending oscillatory phenomena and in keeping track of how
oscilla tions change when a nonlinear functional is applied. Weak
convergence also plays a fundamental role in the modern treatment
of the calculus of variations, again because uniform bounds in norm
for se quences allow to have weak convergent subsequences. In order
to achieve the existence of minimizers for a particular functional,
the property of weak lower semicontinuity should be established
first. This is the crucial and most delicate step in the so-called
direct method of the calculus of variations. A fairly large amount
of work has been devoted to determine under what assumptions we can
have this lower semicontinuity with respect to weak topologies for
nonlin ear functionals in the form of integrals. The conclusion of
all this work is that some type of convexity, understood in a
broader sense, is usually involved."
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