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 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 calculus of variations deals with functionals ${\displaystyle J:V\to {\bar {\mathbb {R} }}}${\displaystyle J:V\to {\bar {\mathbb {R} }}}, where ${\displaystyle V}${\displaystyle V} is some function space and ${\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}}${\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}}. The main interest of the subject is to find minimizers for such functionals, that is, functions ${\displaystyle v\in V}${\displaystyle v\in V} such that ${\displaystyle J(v)\leq J(u)}${\displaystyle J(v)\leq J(u)} for all ${\displaystyle u\in V}${\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 ${\displaystyle J}${\displaystyle J} must be bounded from below to have a minimizer. This means

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

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence ${\displaystyle (u_{n})}${\displaystyle (u_{n})} in ${\displaystyle V}${\displaystyle V} such that ${\displaystyle J(u_{n})\to \inf\{J(u)|u\in V\}.}${\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 ${\displaystyle (u_{n})}${\displaystyle (u_{n})} for ${\displaystyle J}${\displaystyle J}.
2. Show that ${\displaystyle (u_{n})}${\displaystyle (u_{n})} admits some subsequence ${\displaystyle (u_{n_{k}})}${\displaystyle (u_{n_{k}})}, that converges to a ${\displaystyle u_{0}\in V}${\displaystyle u_{0}\in V} with respect to a topology ${\displaystyle \tau }${\displaystyle \tau } on ${\displaystyle V}${\displaystyle V}.
3. Show that ${\displaystyle J}${\displaystyle J} is sequentially lower semi-continuous with respect to the topology ${\displaystyle \tau }${\displaystyle \tau }.

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

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

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

The conclusions follows from

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

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

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

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

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

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

The typical functional in the calculus of variations is an integral of the form

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

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

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

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

As many functionals in the calculus of variations are of the form

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

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

In general one has the following:^[3]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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• Fonseca, Irene; Giovanni Leoni (2007). Modern Methods in the Calculus of Variations: ${\displaystyle L^{p}}${\displaystyle L^{p}} Spaces. Springer. ISBN 978-0-387-35784-3.
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• Calculus of variations

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