direct method in the calculus of variations

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

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

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

This condition 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 $J(u\_n)\; \backslash to\; \backslash inf\backslash \{J(u)|u\backslash in\; V\backslash \}.$

The direct method may be 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 $u\_0\backslash in\; V$ with respect to a topology $\backslash tau$ on $V$.
3. Show that $J$ is sequentially lower semi-continuous with respect to the topology $\backslash tau$.

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

::$\backslash liminf\_\{n\backslash to\backslash infty\}\; J(u\_n)\; \backslash geq\; J(u\_0)$ for any convergent sequence $u\_n\; \backslash to\; u\_0$ in $V$.

The conclusions follows from

:$\backslash inf\backslash \{J(u)|u\backslash in\; V\backslash \}\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; J(u\_n)\; =\; \backslash lim\_\{k\backslash to\; \backslash infty\}\; J(u\_\{n\_k\})\; \backslash geq\; J(u\_0)\; \backslash geq\; \backslash inf\backslash \{J(u)|u\backslash in\; V\backslash \}$,

in other words

:$J(u\_0)\; =\; \backslash inf\backslash \{J(u)|u\backslash in\; V\backslash \}$.

The direct method may often be applied with success when the space $V$ is a subset of a separable reflexive Banach space $W$. In this case the sequential 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 $J:V\backslash to\backslash bar\{\backslash mathbb\{R\}\}$ 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 $u\_n\; \backslash to\; u\_0$ it holds that $\backslash liminf\_\{n\backslash to\backslash infty\}\; J(u\_n)\; \backslash geq\; J(u\_0)$.

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

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

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

:$J(u)\; =\; \backslash int\_\backslash Omega\; F(x,\; u(x),\; \backslash nabla\; u(x))dx$

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

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

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

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

:$J(u)\; =\; \backslash int\_\backslash Omega\; F(x,\; u(x),\; \backslash nabla\; u(x))dx$,

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

In general one has the following:Dacorogna, pp. 74–79.

:Assume that $F$ is a function that has the following properties:

:# The function $F$ is a Carathéodory function.

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

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

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

When $n\; =\; 1$ or $m\; =\; 1$ the following converse-like theorem holdsDacorogna, pp. 66–74.

:Assume that $F$ is continuous and satisfies

::$|\; F(x,\; y,\; A)\; |\; \backslash leq\; a(x,\; |\; y\; |,\; |\; A\; |)$

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

However, there are many interesting cases where one cannot assume that $F$ is convex. The following theoremAcerbi-Fusco proves sequential lower semi-continuity using a weaker notion of convexity:

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

:# The function $F$ is a Carathéodory function.

:# The function $F$ has $p$-growth for some $p>1$: There exists a constant $C$ such that for every $y\; \backslash in\; \backslash mathbb\{R\}^m$ and for almost every $x\; \backslash in\; \backslash Omega$ $|\; F(x,\; y,\; A)\; |\; \backslash leq\; C(1+|y|^p\; +\; |A|^p)$.

:# For every $y\; \backslash in\; \backslash mathbb\{R\}^m$ and for almost every $x\; \backslash in\; \backslash Omega$, the function $A\; \backslash mapsto\; F(x,\; y,\; A)$ is quasiconvex: there exists a cube $D\; \backslash subseteq\; \backslash mathbb\{R\}^n$ such that for every $A\; \backslash in\; \backslash mathbb\{R\}^\{mn\},\; \backslash varphi\; \backslash in\; W^\{1,\backslash infty\}\_0(\backslash Omega,\; \backslash mathbb\{R\}^m)$ it holds:

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

:::where $|D|$ is the volume of $D$.

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

A converse like theorem in this case is the following:

Dacorogna, pp. 156.

:Assume that $F$ is continuous and satisfies

::$|\; F(x,\; y,\; A)\; |\; \backslash leq\; a(x,\; |\; y\; |,\; |\; A\; |)$

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

{{reflist}}

• {{cite book | first = Bernard | last = Dacorogna | year = 1989 | title = Direct Methods in the Calculus of Variations | publisher = Springer-Verlag | isbn = 0-387-50491-5 }}
• {{cite book | first = Irene | last = Fonseca | authorlink = Irene Fonseca |author2=Giovanni Leoni | year = 2007 | title = Modern Methods in the Calculus of Variations: $L^p$ Spaces | publisher = 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}}.
• {{cite news|author=T. Roubíček|title= Direct method for parabolic problems|journal=Adv. Math. Sci. Appl.|volume=10|year=2000|pages=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

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Category:Calculus of variations