Wikipedia / Direct method in the calculus of variations
Wikipedia / 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
is some function space. The main interest of the subject is to find minimizers.
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.
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
The direct method may be broken into the following steps
.
admits some subsequence.
is sequentially lower semi-continuous.
To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
is sequentially lower-semicontinuous if
.
The conclusions follows from
,
in other words
.
Details
Banach spaces
is a subset of a separable reflexive Banach space. In this case the sequential Banach–Alaoglu theorem with respect to the weak topology by showing
is bounded from below,
is bounded, and
.
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
, and its Jacobian-vector.
When deriving the Euler. 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 must then be taken as weak derivatives.
of functions whose trace that satisfy some desired boundary conditions. This is similar to solving the Euler. 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
,
is of great importance.
In general one has the following:
is a function that has the following properties:
is a Carathéodory function.
with Hölder conjugate denotes the Frobenius inner product).
,
is sequentially weakly lower semi-continuous.
the following converse-like theorem holds
is continuous and satisfies
is convex.
is convex.
is convex. The following theorem proves sequential lower semi-continuity using a weaker notion of convexity:
is a function that has the following properties:
is a Carathéodory function.
and for almost every.
and for almost every is quasiconvex it holds:
is the volume.
.
A converse like theorem in this case is the following:
is continuous and satisfies
is quasiconvex, since then quasiconvexity is equivalent to convexity.
Notes
References and further reading
Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin .
Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, .
Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145