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Inmathematics, thedirect method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a givenfunctional,^[1] introduced byStanisław Zaremba andDavid Hilbert around 1900. The method relies on methods offunctional analysis andtopology. 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${\displaystyle J:V\to \overline{\mathbb{R}}}$, where${\displaystyle V}$ is somefunction space and${\displaystyle \overline{\mathbb{R}}=\mathbb{R}\cup \{\mathrm{\infty }\}}$. The main interest of the subject is to findminimizers for such functionals, that is, functions${\displaystyle v\in V}$ such that${\displaystyle J(v)\le J(u)}$ for all${\displaystyle u\in V}$.

The standard tool for obtaining necessary conditions for a function to be a minimizer is theEuler–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

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

This condition is not enough to know that a minimizer exists, but it shows the existence of aminimizing 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 somesubsequence${\displaystyle (u_{n_k})}$, 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 sequentiallylower 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

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}J(u_n)\ge J(u_0)}$ for any convergent sequence${\displaystyle u_n\to u_0}$ in${\displaystyle V}$.

The conclusions follows from

${\displaystyle inf\{J(u)|u\in V\}=\underset{n\to \mathrm{\infty }}{lim}J(u_n)=\underset{k\to \mathrm{\infty }}{lim}J(u_{n_k})\ge J(u_0)\ge inf\{J(u)|u\in V\}}$,

in other words

${\displaystyle J(u_0)=inf\{J(u)|u\in V\}}$.

The direct method may often be applied with success when the space${\displaystyle V}$ is a subset of aseparablereflexiveBanach space${\displaystyle W}$. In this case thesequential 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 theweak 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${\displaystyle J:V\to \overline{\mathbb{R}}}$ 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${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}J(u_n)\ge J(u_0)}$.

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

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

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

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

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

When deriving the Euler–Lagrange equation, the common approach is to assume${\displaystyle \mathrm{\Omega }}$ has a${\displaystyle C^2}$ boundary and let the domain of definition for${\displaystyle J}$ be${\displaystyle C^2(\mathrm{\Omega },{\mathbb{R}}^m)}$. This space is a Banach space when endowed with thesupremum norm, but it is not reflexive. When applying the direct method, the functional is usually defined on aSobolev space${\displaystyle W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$ 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 asweak derivatives.

Another common function space is${\displaystyle W_g^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$ which is the affine sub space of${\displaystyle W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$ of functions whosetrace 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${\displaystyle W_g^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$ but not in${\displaystyle W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$.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

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

where${\displaystyle \mathrm{\Omega }\subseteq {\mathbb{R}}^n}$ is open, theorems characterizing functions${\displaystyle F}$ for which${\displaystyle J}$ is weakly sequentially lower-semicontinuous in${\displaystyle W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^m)}$ with${\displaystyle p\ge 1}$ 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 aCarathéodory function.
2. There exist${\displaystyle a\in L^q(\mathrm{\Omega },{\mathbb{R}}^{mn})}$ withHölder conjugate${\displaystyle q=\frac{p}{p-1}}$ and${\displaystyle b\in L^1(\mathrm{\Omega })}$ such that the following inequality holds true for almost every${\displaystyle x\in \mathrm{\Omega }}$ and every${\displaystyle (y,A)\in {\mathbb{R}}^m\times {\mathbb{R}}^{mn}}$:${\displaystyle F(x,y,A)\ge ⟨a(x),A⟩+b(x)}$. Here,${\displaystyle ⟨a(x),A⟩}$ denotes theFrobenius 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${\displaystyle x\in \mathrm{\Omega }}$ 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

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

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${\displaystyle (x,y)\in \mathrm{\Omega }\times {\mathbb{R}}^m}$ 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${\displaystyle F:\mathrm{\Omega }\times {\mathbb{R}}^m\times {\mathbb{R}}^{mn}\to [0,\mathrm{\infty })}$ is a function that has the following properties:

1. The function${\displaystyle F}$ is aCarathéodory 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 foralmost every${\displaystyle x\in \mathrm{\Omega }}$${\displaystyle |F(x,y,A)|\le C(1+|y{|}^p+|A{|}^p)}$.
3. For every${\displaystyle y\in {\mathbb{R}}^m}$ and foralmost every${\displaystyle x\in \mathrm{\Omega }}$, the function${\displaystyle A\mapsto F(x,y,A)}$ isquasiconvex: there exists a cube${\displaystyle D\subseteq {\mathbb{R}}^n}$ such that for every${\displaystyle A\in {\mathbb{R}}^{mn},\phi \in W_0^{1,\mathrm{\infty }}(\mathrm{\Omega },{\mathbb{R}}^m)}$ it holds:

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

where${\displaystyle |D|}$ is thevolume of${\displaystyle D}$.

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

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

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

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

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${\displaystyle (x,y)\in \mathrm{\Omega }\times {\mathbb{R}}^m}$ the function${\displaystyle A\mapsto F(x,y,A)}$ isquasiconvex. 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.

• 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), BerlinISBN 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

• Calculus of variations

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