Direct method in calculus of variations
Encyclopedia
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.
, 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
It 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
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
.
is a subset of a reflexive
Banach space
. In this case the 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
The second part is usually accomplished by showing that
admits some growth condition. An example is
for some 
, 
and 
.
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.
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 spaces. The derivatives of 
in the formula for 
must then be taken as weak derivative
s. The next section presents two theorems regarding weak sequential lower semi-continuity of functionals of the above type.
,
where
is open, theorems characterizing functions 
for which 
is weakly sequentially lower-semicontinuous in 
is of great importance.
In general we have the following
When
or 
the following converse-like theorem holds
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.
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, a topic in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
, introduced by Zaremba and David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
around 1900. The method relies on methods of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. 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 is some function spaceFunction space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
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 meansIt 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 subsequenceSubsequenceIn mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
, 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 function
is sequentially lower-semicontinuous if
for any convergent sequence 
in 
.
The conclusions follows from
,in other words
.Banach spaces
The direct method may often be applied with success when the space is a subset of a reflexiveReflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. In this case the Banach–Alaoglu theorem implies, that any bounded sequence in has a subsequence that converges to some in with respect to the weak topologyWeak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
. 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 for some , and .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 formwhere
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 spaceSobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
with , which is a reflexive Banach spaces. The derivatives of in the formula for must then be taken as weak derivativeWeak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...
s. 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 function
- 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 polyconvexityPolyconvex function
In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×n denote the space of all m × n matrices over the field K, which may be either the real numbers R, or the complex numbers C...
and quasiconvexity.