Direct method in the calculus of variations
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 formula_1, where formula_2 is some function space and formula_3. The main interest of  the subject is to find "minimizers" for such functionals, that is, functions formula_4 such that:formula_5 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 formula_6 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 formula_8 in formula_2 such  that formula_10 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 formula_2 is a subset of a separable reflexive Banach space formula_27. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence formula_8 in formula_2 has a subsequence that converges to some formula_30 in formula_27 with respect to the weak topology. If formula_2 is sequentially  closed in formula_27, so that formula_30 is in formula_2, the direct method may be applied to a functional formula_36 by showing The second part is usually accomplished by showing that formula_6 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 formula_48 is a subset of formula_49 and formula_50 is  a real-valued function on formula_51. The argument of formula_6 is a differentiable function formula_53, and its Jacobian formula_54 is identified with a formula_55-vector. When deriving the Euler–Lagrange equation, the common approach is to assume formula_48 has a formula_57 boundary and let the domain of definition for formula_6 be formula_59. 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 formula_60 with formula_61, which is a reflexive Banach space. The derivatives of formula_62 in the formula for  formula_6 must then be taken as weak derivatives. The next section presents two 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 formula_65 is open, theorems characterizing functions formula_50 for which formula_6 is weakly sequentially lower-semicontinuous in formula_60 with formula_69 is of great importance. In general one has the following: When formula_89 or formula_90 the following converse-like theorem holds In conclusion, when formula_90 or formula_89, the functional formula_6, assuming reasonable growth and boundedness on formula_50 , is weakly sequentially lower semi-continuous if, and only if the function formula_85 is convex. If both formula_106 and formula_107 are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely polyconvexity and quasiconvexity.