Direct method in the calculus of variations

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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 ${\displaystyle J(v)\leq J(u)}$ for all ${\displaystyle u\in V}$.

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.

Notes

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. Vol. 10. pp. 57–65. MR 1769181.
• Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145