Direct method in the calculus of variations

In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by 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 ${\displaystyle J:V\to {\bar {\mathbb {R} }}}$, where ${\displaystyle V}$ is some function space and ${\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}}$. The main interest of the subject is to find minimizers for such functionals, that is, functions ${\displaystyle v\in V}$ such that:${\displaystyle J(v)\leq J(u)\forall u\in V.}$

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