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Concept
Direct method in the calculus of variations
Summary
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.
The calculus of variations deals with functionals , where is some function space and . The main interest of the subject is to find minimizers for such functionals, that is, functions such that:
The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand.
The functional must be bounded from below to have a minimizer. This means
This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence in such that
The direct method may be broken into the following steps
Take a minimizing sequence for .
Show that admits some subsequence , that converges to a with respect to a topology on .
Show that is sequentially lower semi-continuous with respect to the topology .
To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
The function is sequentially lower-semicontinuous if
for any convergent sequence in .
The conclusions follows from
in other words
The direct method may often be applied with success when the space is a subset of a separable reflexive Banach space . In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence in has a subsequence that converges to some in with respect to the weak topology. If is sequentially closed in , so that is in , the direct method may be applied to a functional by showing
is bounded from below,
any minimizing sequence for is bounded, and
is weakly sequentially lower semi-continuous, i.
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