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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
around 1900. The method relies on methods of
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined on ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
. 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 space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vec ...
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 be broken into the following steps
# Take a minimizing sequence
for
.
# Show that
admits some
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is ...
, that converges to a
with respect to a topology
on
.
# Show that
is sequentially
lower semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...
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
:
.
Details
Banach spaces
The direct method may often be applied with success when the space
is a subset of a
separable reflexive Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
. 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
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
. 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 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
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
, but it is not reflexive. When applying the direct method, the functional is usually defined on a
Sobolev 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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
with
, which is a reflexive Banach space. The derivatives of
in the formula for
must then be taken as
weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b.
The method ...
s.
Another common function space is
which is the affine sub space of
of functions whose
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
is some fixed function
in the image of the trace operator. This restriction allows finding minimizers of the functional
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
for which
is weakly sequentially lower-semicontinuous in
with
is of great importance.
In general one has the following:
:Assume that
is a function that has the following properties:
:# The function
is a
Carathéodory function In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue- ...
.
:# There exist
with
Hölder conjugate In mathematics, two real numbers p, q>1 are called conjugate indices (or Hölder conjugates) if
: \frac + \frac = 1.
Formally, we also define q = \infty as conjugate to p=1 and vice versa
References
Additional references
*
*
{{Lat ...
and
such that the following inequality holds true for almost every
and every
:
. Here,
denotes the
Frobenius inner product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though t ...
of
and
in
).
:If the function
is convex for almost every
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
. 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.
However, there are many interesting cases where one cannot assume that
is convex. The following theorem proves sequential lower semi-continuity using a weaker notion of convexity:
:Assume that
is a function that has the following properties:
:# The function
is a
Carathéodory function In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue- ...
.
:# The function
has
-growth for some
: There exists a constant
such that for every
and for almost every
.
:# For every
and for almost every
, the function
is
quasiconvex
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single ...
: there exists a cube
such that for every
it holds:
:::where
is the
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...
of
.
:Then
is sequentially weakly lower semi-continuous in
.
A converse like theorem in this case is the following:
[Dacorogna, pp. 156.]
:Assume that
is continuous and satisfies
::
:for every
, and a fixed function
increasing in
and
, and locally integrable in
. If
is sequentially weakly lower semi-continuous, then for any given
the function
is
quasiconvex
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single ...
. The claim is true even when both
are bigger than
and coincides with the previous claim when
or
, 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
{{DEFAULTSORT:Direct Method In The Calculus Of Variations
Calculus of variations