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,[1] introduced by Stanisław Zaremba and David Hilbert around 1900.
As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.
[2] The calculus of variations deals with functionals
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.
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
The direct method may be broken into the following steps To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
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
with respect to the weak topology.
, the direct method may be applied to a functional
An example is A functional with this property is sometimes called coercive.
Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method.
See below for some theorems for a general class of functionals.
The typical functional in the calculus of variations is an integral of the form where
When deriving the Euler–Lagrange equation, the common approach is to assume
When applying the direct method, the functional is usually defined on a Sobolev space
This restriction allows finding minimizers of the functional
that satisfy some desired boundary conditions.
This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions.
The idea of solving minimization problems while restricting the values on the boundary can be further generalized by looking on function spaces where the trace is fixed only on a part of the boundary, and can be arbitrary on the rest.
The next section presents theorems regarding weak sequential lower semi-continuity of functionals of the above type.
As many functionals in the calculus of variations are of the form where
is open, theorems characterizing functions
is weakly sequentially lower-semicontinuous in
the following converse-like theorem holds[4] In conclusion, when
, assuming reasonable growth and boundedness on
, is weakly sequentially lower semi-continuous if, and only if the function
The following theorem[5] proves sequential lower semi-continuity using a weaker notion of convexity: