User:Niels1004/Direct method (Calculus of variations)

In the calculus of variations, a topic in mathematics, the direct method is a way of attacking the problem of minimizing (or maximizing) a real-valued functional by methods of topology and functional analysis. The direct method deals directly with the functional in question, in constrast to the more classical approach of studying the Euler-Lagrange equation of the functional.

Introduction
The calculus of variations deals with functionals $$J:X\to\mathbb{R}\cup\{\infty\}$$, where $$X$$ is some function space. The typical objective is to minimize such a functional on a fixed subset $$Y \subset X$$. Assuming $$J$$ is bounded from below, let
 * $$m = \inf\{J(g)|g\in Y\} > -\infty $$.

Then there exists a sequence $$(f_n)$$ in $$Y$$ such that $$J(f_n) \to m$$. This is called a minimizing sequence. The direct method is to equip $$X$$ with a topology that will guarantee that the minimizing sequence $$(f_n)$$ has a subsequence that converges to a function $$g \in Y$$, and with respect to which $$J$$ has enough continuity to assure $$J(g) = m$$. This is not always possible, but if is, it will show that $$J$$ attains its infimum on $$Y$$ and thus have a minimum.

Abstract example
The direct method may often be applied with success to functionals defined on a reflexive Banach space. For such a space $$X$$ the Banach-Alaoglu theorem implies, that any bounded sequence has a subsequence which converges in the weak topology on $$X$$. For a functional $$J:X\to\mathbb{R}\cup\{\infty\}$$ the direct method may then be applied by showing The second part is usually accomplished by showing that $$J$$ admits some growth condition. An example is
 * 1) $$J$$ is bounded from below,
 * 2) any minimizing sequence for $$J$$ is bounded, and
 * 3) $$J$$ is sequentially weakly lower semi-continuous i.e., for any weakly convergent sequence $$x_n \to y$$ it holds that $$\liminf_{n\to\infty} J(x_n) \geq J(y)$$.
 * $$J(x) \geq \alpha \lVert x \rVert^q - \beta$$ for some $$\alpha > 0$$, $$q \geq 1$$ and $$\beta \geq 0$$.

A functional with this property is sometimes called coercive.

Calculus of variations
The typical functional in the calculus of variations is an integral of the form
 * $$J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx$$

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

When deriving the Euler-Lagrange equation, the common approach is to assume $$\Omega$$ has a $$C^2$$ boundary and let the domain of definition for $$J$$ be $$C^2(\Omega, \mathbb{R}^m)$$. This space is a Banach space when endowed with the supremum norm, but it is not reflexive. A better space of definition in the context of the direct method is a Sobolev space $$W^{1,p}(\Omega, \mathbb{R}^m)$$ with $$p > 1$$, which is reflexive Banach spaces. The derivatives of $$u$$ in the formula for $$J$$ must be then be taken as weak derivatives.

There are various theorems that characterize the functions $$F$$ for which $$J$$ is sequentially weakly lower semi-continuous. In general dimensions we have the following
 * Suppose $$F$$ is continuously differentiable and bounded from below. If for any $$x\in \Omega$$ and $$y\in \mathbb{R}^m$$, the map $$p \mapsto F(x, y, p)$$ is convex, then $$J$$ is sequentially weakly lower semi-continuous.