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. The method relies on methods of functional analysis and topology.

The method

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

$J(v)\leq J(u)$ for any $u\in V$.

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 $J$ must be bounded from below to have a minimizer. This means

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

It is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence $(u_{n})$ in $V$ such that

$J(u_{n})\to \inf\{J(u)|u\in V\}$.

The direct method may broken into the following steps

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

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

The function $J$ is sequentially lower-semicontinuous if

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

The conclusions follows from

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

in other words

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

Details

Banach spaces

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

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

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

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

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

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

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

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

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

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

In general we have the following

Assume that $F$ is a function such that

1. The function $(y,p)\mapsto F(x,y,p)$ is continuous for almost every $x\in \Omega$,
2. the function $x\mapsto F(x,y,p)$ is measurable for every $(y,p)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn}$, and
3. $F(x,y,p)\geq a(x)\cdot p+b(x)$ for a fixed $a\in L^{q}(\Omega ,\mathbb {R} ^{m})$ where $1/q+1/p=1$, a fixed $b\in L^{1}(\Omega )$, for a.e. $x\in \Omega$ and every $(y,p)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn}$ (here $a(x)\cdot p$ means the inner product of $a(x)$ and $p$ in $\mathbb {R} ^{mn}$).

The following holds. If the function $p\mapsto F(x,y,p)$ is convex for a.e. $x\in \Omega$ and every $y\in \mathbb {R} ^{m}$,

then $J$ is sequentially weakly lower semi-continuous.

When $n=1$ or $m=1$ the following converse-like theorem holds

Assume that $F$ is continuous and satisfies

$|F(x,y,p)|\leq a(x,|y|,|p|)$

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

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

• Dacorogna, Bernard (1989). Direct Methods in the Calculus of Variations. Springer-Verlag. ISBN 0-387-50491-5.
• Fonseca, Irene (2007). Modern Methods in the Calculus of Variations: $L^{p}$ Spaces. Springer. ISBN 978-0-387-35784-3. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)