Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

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,^[1] 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.^[2]

The method

The calculus of variations deals with functionals ${\displaystyle J:V\to {\bar {\mathbb {R} ))}$, where ${\displaystyle V}$ is some function space and ${\displaystyle {\bar {\mathbb {R} ))=\mathbb {R} \cup \{\infty \))$. The main interest of the subject is to find minimizers for such functionals, that is, functions ${\displaystyle v\in V}$ such that:${\displaystyle J(v)\leq J(u)\forall u\in V.}$

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 ${\displaystyle J}$ must be bounded from below to have a minimizer. This means

${\displaystyle \inf\{J(u)|u\in V\}>-\infty .\,}$

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence ${\displaystyle (u_{n})}$ in ${\displaystyle V}$ such that ${\displaystyle J(u_{n})\to \inf\{J(u)|u\in V\}.}$

The direct method may be broken into the following steps

1. Take a minimizing sequence ${\displaystyle (u_{n})}$ for ${\displaystyle J}$.
2. Show that ${\displaystyle (u_{n})}$ admits some subsequence ${\displaystyle (u_{n_{k)))}$, that converges to a ${\displaystyle u_{0}\in V}$ with respect to a topology ${\displaystyle \tau }$ on ${\displaystyle V}$.
3. Show that ${\displaystyle J}$ is sequentially lower semi-continuous with respect to the topology ${\displaystyle \tau }$.

To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.

The function ${\displaystyle J}$ is sequentially lower-semicontinuous if

${\displaystyle \liminf _{n\to \infty }J(u_{n})\geq J(u_{0})}$ for any convergent sequence ${\displaystyle u_{n}\to u_{0))$ in ${\displaystyle V}$.

The conclusions follows from

${\displaystyle \inf\{J(u)|u\in V\}=\lim _{n\to \infty }J(u_{n})=\lim _{k\to \infty }J(u_{n_{k)))\geq J(u_{0})\geq \inf\{J(u)|u\in V\))$,

in other words

${\displaystyle J(u_{0})=\inf\{J(u)|u\in V\))$.

Details

Banach spaces

The direct method may often be applied with success when the space ${\displaystyle V}$ is a subset of a separable reflexive Banach space ${\displaystyle W}$. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence ${\displaystyle (u_{n})}$ in ${\displaystyle V}$ has a subsequence that converges to some ${\displaystyle u_{0))$ in ${\displaystyle W}$ with respect to the weak topology. If ${\displaystyle V}$ is sequentially closed in ${\displaystyle W}$, so that ${\displaystyle u_{0))$ is in ${\displaystyle V}$, the direct method may be applied to a functional ${\displaystyle J:V\to {\bar {\mathbb {R} ))}$ by showing

1. ${\displaystyle J}$ is bounded from below,
2. any minimizing sequence for ${\displaystyle J}$ is bounded, and
3. ${\displaystyle J}$ is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence ${\displaystyle u_{n}\to u_{0))$ it holds that ${\displaystyle \liminf _{n\to \infty }J(u_{n})\geq J(u_{0})}$.

The second part is usually accomplished by showing that ${\displaystyle J}$ admits some growth condition. An example is

${\displaystyle J(x)\geq \alpha \lVert x\rVert ^{q}-\beta }$ for some ${\displaystyle \alpha >0}$, ${\displaystyle q\geq 1}$ and ${\displaystyle \beta \geq 0}$.

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

${\displaystyle J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx}$

where ${\displaystyle \Omega }$ is a subset of ${\displaystyle \mathbb {R} ^{n))$ and ${\displaystyle F}$ is a real-valued function on ${\displaystyle \Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn))$. The argument of ${\displaystyle J}$ is a differentiable function ${\displaystyle u:\Omega \to \mathbb {R} ^{m))$, and its Jacobian ${\displaystyle \nabla u(x)}$ is identified with a ${\displaystyle mn}$-vector.

When deriving the Euler–Lagrange equation, the common approach is to assume ${\displaystyle \Omega }$ has a ${\displaystyle C^{2))$ boundary and let the domain of definition for ${\displaystyle J}$ be ${\displaystyle C^{2}(\Omega ,\mathbb {R} ^{m})}$. 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 ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}$ with ${\displaystyle p>1}$, which is a reflexive Banach space. The derivatives of ${\displaystyle u}$ in the formula for ${\displaystyle J}$ must then be taken as weak derivatives.

Another common function space is ${\displaystyle W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})}$ which is the affine sub space of ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}$ of functions whose trace is some fixed function ${\displaystyle g}$ in the image of the trace operator. This restriction allows finding minimizers of the functional ${\displaystyle J}$ 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 ${\displaystyle W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})}$ but not in ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}$. 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

${\displaystyle J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx}$,

where ${\displaystyle \Omega \subseteq \mathbb {R} ^{n))$ is open, theorems characterizing functions ${\displaystyle F}$ for which ${\displaystyle J}$ is weakly sequentially lower-semicontinuous in ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}$ with ${\displaystyle p\geq 1}$ is of great importance.

In general one has the following:^[3]

Assume that ${\displaystyle F}$ is a function that has the following properties:

1. The function ${\displaystyle F}$ is a Carathéodory function.
2. There exist ${\displaystyle a\in L^{q}(\Omega ,\mathbb {R} ^{mn})}$ with Hölder conjugate ${\displaystyle q={\tfrac {p}{p-1))}$ and ${\displaystyle b\in L^{1}(\Omega )}$ such that the following inequality holds true for almost every ${\displaystyle x\in \Omega }$ and every ${\displaystyle (y,A)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn))$: ${\displaystyle F(x,y,A)\geq \langle a(x),A\rangle +b(x)}$. Here, ${\displaystyle \langle a(x),A\rangle }$ denotes the Frobenius inner product of ${\displaystyle a(x)}$ and ${\displaystyle A}$ in ${\displaystyle \mathbb {R} ^{mn))$).

If the function ${\displaystyle A\mapsto F(x,y,A)}$ is convex for almost every ${\displaystyle x\in \Omega }$ and every ${\displaystyle y\in \mathbb {R} ^{m))$,

then ${\displaystyle J}$ is sequentially weakly lower semi-continuous.

When ${\displaystyle n=1}$ or ${\displaystyle m=1}$ the following converse-like theorem holds^[4]

Assume that ${\displaystyle F}$ is continuous and satisfies

${\displaystyle |F(x,y,A)|\leq a(x,|y|,|A|)}$

for every ${\displaystyle (x,y,A)}$, and a fixed function ${\displaystyle a(x,|y|,|A|)}$ increasing in ${\displaystyle |y|}$ and ${\displaystyle |A|}$, and locally integrable in ${\displaystyle x}$. If ${\displaystyle J}$ is sequentially weakly lower semi-continuous, then for any given ${\displaystyle (x,y)\in \Omega \times \mathbb {R} ^{m))$ the function ${\displaystyle A\mapsto F(x,y,A)}$ is convex.

In conclusion, when ${\displaystyle m=1}$ or ${\displaystyle n=1}$, the functional ${\displaystyle J}$, assuming reasonable growth and boundedness on ${\displaystyle F}$, is weakly sequentially lower semi-continuous if, and only if the function ${\displaystyle A\mapsto F(x,y,A)}$ is convex.

However, there are many interesting cases where one cannot assume that ${\displaystyle F}$ is convex. The following theorem^[5] proves sequential lower semi-continuity using a weaker notion of convexity:

Assume that ${\displaystyle F:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn}\to [0,\infty )}$ is a function that has the following properties:

1. The function ${\displaystyle F}$ is a Carathéodory function.
2. The function ${\displaystyle F}$ has ${\displaystyle p}$-growth for some ${\displaystyle p>1}$: There exists a constant ${\displaystyle C}$ such that for every ${\displaystyle y\in \mathbb {R} ^{m))$ and for almost every ${\displaystyle x\in \Omega }$ ${\displaystyle |F(x,y,A)|\leq C(1+|y|^{p}+|A|^{p})}$.
3. For every ${\displaystyle y\in \mathbb {R} ^{m))$ and for almost every ${\displaystyle x\in \Omega }$, the function ${\displaystyle A\mapsto F(x,y,A)}$ is quasiconvex: there exists a cube ${\displaystyle D\subseteq \mathbb {R} ^{n))$ such that for every ${\displaystyle A\in \mathbb {R} ^{mn},\varphi \in W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{m})}$ it holds:

$${\displaystyle F(x,y,A)\leq |D|^{-1}\int _{D}F(x,y,A+\nabla \varphi (z))dz}$$

where ${\displaystyle |D|}$ is the volume of ${\displaystyle D}$.

Then ${\displaystyle J}$ is sequentially weakly lower semi-continuous in ${\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}$.

A converse like theorem in this case is the following: ^[6]

Assume that ${\displaystyle F}$ is continuous and satisfies

${\displaystyle |F(x,y,A)|\leq a(x,|y|,|A|)}$

for every ${\displaystyle (x,y,A)}$, and a fixed function ${\displaystyle a(x,|y|,|A|)}$ increasing in ${\displaystyle |y|}$ and ${\displaystyle |A|}$, and locally integrable in ${\displaystyle x}$. If ${\displaystyle J}$ is sequentially weakly lower semi-continuous, then for any given ${\displaystyle (x,y)\in \Omega \times \mathbb {R} ^{m))$ the function ${\displaystyle A\mapsto F(x,y,A)}$ is quasiconvex. The claim is true even when both ${\displaystyle m,n}$ are bigger than ${\displaystyle 1}$ and coincides with the previous claim when ${\displaystyle m=1}$ or ${\displaystyle n=1}$, since then quasiconvexity is equivalent to convexity.