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 , 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 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 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.e., for any weakly convergent sequence it holds that .
The second part is usually accomplished by showing that 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.
The typical functional in the calculus of variations is an integral of the form
where is a subset of and is a real-valued function on . The argument of is a differentiable function , and its Jacobian is identified with a -vector.
When deriving the Euler–Lagrange equation, the common approach is to assume has a boundary and let the domain of definition for 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 , which is a reflexive Banach space. The derivatives of in the formula for 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 is open, theorems characterizing functions for which is weakly sequentially lower-semicontinuous in is of great importance.
In general we have the following
- Assume that is a function such that
- The function is continuous for almost every ,
- the function is measurable for every , and
- for a fixed where , a fixed , for a.e. and every (here means the inner product of and in ).
- The following holds. If the function is convex for a.e. and every ,
- then is sequentially weakly lower semi-continuous.
When or the following converse-like theorem holds
- Assume that is continuous and satisfies
- for every , and a fixed function increasing in and , and locally integrable in . It then holds, if is sequentially weakly lower semi-continuous, then for any given the function is convex.
In conclusion, when or , the functional , assuming reasonable growth and boundedness on , is weakly sequentially lower semi-continuous if, and only if, the function is convex. If both and are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely polyconvexity and quasiconvexity.
References and further reading
- Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, ISBN 978-3-642-10455-8.
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