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 method

The calculus of variations deals with functionals, where V is some function space and. The main interest of the subject is to find minimizers for such functionals, that is, functions v \in V such that for all u \in V. 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 J 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 (u_n) in V such that 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

Details

Banach spaces

The direct method may often be applied with success when the space V is a subset of a separable reflexive Banach space W. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence (u_n) in V has a subsequence that converges to some u_0 in W with respect to the weak topology. If V is sequentially closed in W, so that u_0 is in V, the direct method may be applied to a functional by showing The second part is usually accomplished by showing that J admits some growth condition. 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.

Sobolev spaces

The typical functional in the calculus of variations is an integral of the form where \Omega is a subset of and F is a real-valued function on. The argument of J is a differentiable function, and its Jacobian \nabla u(x) is identified with a mn-vector. When deriving the Euler–Lagrange equation, the common approach is to assume \Omega has a C^2 boundary and let the domain of definition for J be. This space is a Banach space when endowed with the supremum norm, but it is not reflexive. When applying the direct method, the functional is usually defined on a Sobolev space with p > 1, which is a reflexive Banach space. The derivatives of u in the formula for J must then be taken as weak derivatives. Another common function space is which is the affine sub space of of functions whose trace is some fixed function g in the image of the trace operator. This restriction allows finding minimizers of the functional J that satisfy some desired boundary conditions. This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions. Additionally there are settings in which there are minimizers in but not in. 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.

Sequential lower semi-continuity of integrals

As many functionals in the calculus of variations are of the form where is open, theorems characterizing functions F for which J is weakly sequentially lower-semicontinuous in with p \geq 1 is of great importance. In general one has the following: When n = 1 or m = 1 the following converse-like theorem holds In conclusion, when m = 1 or n = 1, the functional J, assuming reasonable growth and boundedness on F, is weakly sequentially lower semi-continuous if, and only if the function is convex. However, there are many interesting cases where one cannot assume that F is convex. The following theorem proves sequential lower semi-continuity using a weaker notion of convexity: A converse like theorem in this case is the following:

References and further reading