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,[1] 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.[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 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. 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

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

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 ${\displaystyle (y,A)\mapsto F(x,y,A)}$ is continuous for almost every ${\displaystyle x\in \Omega }$.
2. The function ${\displaystyle x\mapsto F(x,y,A)}$ is measurable for every ${\displaystyle (y,A)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn}}$.
3. 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 ${\displaystyle p}$, 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.

If both ${\displaystyle n}$ and ${\displaystyle m}$ are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely polyconvexity and quasiconvexity.[5]

Notes

1. Dacorogna, pp. 1–43.
2. I. M. Gelfand; S. V. Fomin (1991). Calculus of Variations. Dover Publications. ISBN 978-0-486-41448-5.
3. Dacorogna, pp. 74–79.
4. Dacorogna, pp. 66–74.
5. Dacorogna, pp. 87–185.

References and further reading

• Dacorogna, Bernard (1989). Direct Methods in the Calculus of Variations. Springer-Verlag. ISBN 0-387-50491-5.
• Fonseca, Irene; Giovanni Leoni (2007). Modern Methods in the Calculus of Variations: ${\displaystyle L^{p}}$ Spaces. Springer. ISBN 978-0-387-35784-3.
• Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin ISBN 978-3-540-69915-6.
• Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, ISBN 978-3-642-10455-8.
• T. Roubíček (2000). "Direct method for parabolic problems". Adv. Math. Sci. Appl. 10. pp. 57–65. MR 1769181.