Английская Википедия:Direct method in the calculus of variations
Шаблон:Short description Шаблон:Calculus 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 <math>J:V \to \bar{\mathbb{R}}</math>, where <math>V</math> is some function space and <math>\bar{\mathbb{R}} = \mathbb{R} \cup \{\infty\}</math>. The main interest of the subject is to find minimizers for such functionals, that is, functions <math>v \in V</math> such that:<math>J(v) \leq J(u)\forall u \in V. </math>
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 <math>J</math> must be bounded from below to have a minimizer. This means
- <math>\inf\{J(u)|u\in V\} > -\infty.\,</math>
This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence <math>(u_n)</math> in <math>V</math> such that <math>J(u_n) \to \inf\{J(u)|u\in V\}.</math>
The direct method may be broken into the following steps
- Take a minimizing sequence <math>(u_n)</math> for <math>J</math>.
- Show that <math>(u_n)</math> admits some subsequence <math>(u_{n_k})</math>, that converges to a <math>u_0\in V</math> with respect to a topology <math>\tau</math> on <math>V</math>.
- Show that <math>J</math> is sequentially lower semi-continuous with respect to the topology <math>\tau</math>.
To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
- The function <math>J</math> is sequentially lower-semicontinuous if
- <math>\liminf_{n\to\infty} J(u_n) \geq J(u_0)</math> for any convergent sequence <math>u_n \to u_0</math> in <math>V</math>.
The conclusions follows from
- <math>\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\}</math>,
in other words
- <math>J(u_0) = \inf\{J(u)|u\in V\}</math>.
Details
Banach spaces
The direct method may often be applied with success when the space <math>V</math> is a subset of a separable reflexive Banach space <math>W</math>. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence <math>(u_n)</math> in <math>V</math> has a subsequence that converges to some <math>u_0</math> in <math>W</math> with respect to the weak topology. If <math>V</math> is sequentially closed in <math>W</math>, so that <math>u_0</math> is in <math>V</math>, the direct method may be applied to a functional <math>J:V\to\bar{\mathbb{R}}</math> by showing
- <math>J</math> is bounded from below,
- any minimizing sequence for <math>J</math> is bounded, and
- <math>J</math> is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence <math>u_n \to u_0</math> it holds that <math>\liminf_{n\to\infty} J(u_n) \geq J(u_0)</math>.
The second part is usually accomplished by showing that <math>J</math> admits some growth condition. An example is
- <math>J(x) \geq \alpha \lVert x \rVert^q - \beta</math> for some <math>\alpha > 0</math>, <math>q \geq 1</math> and <math>\beta \geq 0</math>.
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
- <math>J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx</math>
where <math>\Omega</math> is a subset of <math>\mathbb{R}^n</math> and <math>F</math> is a real-valued function on <math>\Omega \times \mathbb{R}^m \times \mathbb{R}^{mn}</math>. The argument of <math>J</math> is a differentiable function <math>u:\Omega \to \mathbb{R}^m</math>, and its Jacobian <math>\nabla u(x)</math> is identified with a <math>mn</math>-vector.
When deriving the Euler–Lagrange equation, the common approach is to assume <math>\Omega</math> has a <math>C^2</math> boundary and let the domain of definition for <math>J</math> be <math>C^2(\Omega, \mathbb{R}^m)</math>. 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 <math>W^{1,p}(\Omega, \mathbb{R}^m)</math> with <math>p > 1</math>, which is a reflexive Banach space. The derivatives of <math>u</math> in the formula for <math>J</math> must then be taken as weak derivatives.
Another common function space is <math>W^{1,p}_g(\Omega, \mathbb{R}^m)</math> which is the affine sub space of <math>W^{1,p}(\Omega, \mathbb{R}^m)</math> of functions whose trace is some fixed function <math>g</math> in the image of the trace operator. This restriction allows finding minimizers of the functional <math>J</math> 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 <math>W^{1,p}_g(\Omega, \mathbb{R}^m)</math> but not in <math>W^{1,p}(\Omega, \mathbb{R}^m)</math>. 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
- <math>J(u) = \int_\Omega F(x, u(x), \nabla u(x))dx</math>,
where <math>\Omega \subseteq \mathbb{R}^n</math> is open, theorems characterizing functions <math>F</math> for which <math>J</math> is weakly sequentially lower-semicontinuous in <math>W^{1,p}(\Omega, \mathbb{R}^m)</math> with <math>p \geq 1</math> is of great importance.
In general one has the following:[3]
- Assume that <math>F</math> is a function that has the following properties:
- The function <math>F</math> is a Carathéodory function.
- There exist <math>a\in L^q(\Omega, \mathbb{R}^{mn})</math> with Hölder conjugate <math>q = \tfrac{p}{p-1}</math> and <math>b \in L^1(\Omega)</math> such that the following inequality holds true for almost every <math>x \in \Omega</math> and every <math>(y, A) \in \mathbb{R}^m \times \mathbb{R}^{mn}</math>: <math>F(x, y, A) \geq \langle a(x) , A \rangle + b(x)</math>. Here, <math>\langle a(x) , A \rangle</math> denotes the Frobenius inner product of <math>a(x)</math> and <math>A</math> in <math>\mathbb{R}^{mn}</math>).
- If the function <math>A \mapsto F(x, y, A)</math> is convex for almost every <math>x \in \Omega</math> and every <math>y\in \mathbb{R}^m</math>,
- then <math>J</math> is sequentially weakly lower semi-continuous.
When <math>n = 1</math> or <math>m = 1</math> the following converse-like theorem holds[4]
- Assume that <math>F</math> is continuous and satisfies
- <math>| F(x, y, A) | \leq a(x, | y |, | A |)</math>
- for every <math>(x, y, A)</math>, and a fixed function <math>a(x, |y|, |A|)</math> increasing in <math>|y|</math> and <math>|A|</math>, and locally integrable in <math>x</math>. If <math>J</math> is sequentially weakly lower semi-continuous, then for any given <math>(x, y) \in \Omega \times \mathbb{R}^m</math> the function <math>A \mapsto F(x, y, A)</math> is convex.
In conclusion, when <math>m = 1</math> or <math>n = 1</math>, the functional <math>J</math>, assuming reasonable growth and boundedness on <math>F</math>, is weakly sequentially lower semi-continuous if, and only if the function <math>A \mapsto F(x, y, A)</math> is convex.
However, there are many interesting cases where one cannot assume that <math>F</math> is convex. The following theorem[5] proves sequential lower semi-continuity using a weaker notion of convexity:
- Assume that <math>F: \Omega \times \mathbb{R}^m \times \mathbb{R}^{mn} \to [0, \infty)</math> is a function that has the following properties:
- The function <math>F</math> is a Carathéodory function.
- The function <math>F</math> has <math>p</math>-growth for some <math>p>1</math>: There exists a constant <math>C</math> such that for every <math>y \in \mathbb{R}^m</math> and for almost every <math>x \in \Omega</math> <math>| F(x, y, A) | \leq C(1+|y|^p + |A|^p)</math>.
- For every <math>y \in \mathbb{R}^m</math> and for almost every <math>x \in \Omega</math>, the function <math>A \mapsto F(x, y, A) </math> is quasiconvex: there exists a cube <math>D \subseteq \mathbb{R}^n</math> such that for every <math>A \in \mathbb{R}^{mn}, \varphi \in W^{1,\infty}_0(\Omega, \mathbb{R}^m)</math> it holds:
<math display=block> F(x, y, A) \leq |D|^{-1} \int_D F(x, y, A+ \nabla \varphi (z))dz </math>
- where <math>|D|</math> is the volume of <math>D</math>.
- Then <math>J</math> is sequentially weakly lower semi-continuous in <math> W^{1,p}(\Omega,\mathbb{R}^m) </math>.
A converse like theorem in this case is the following: [6]
- Assume that <math>F</math> is continuous and satisfies
- <math>| F(x, y, A) | \leq a(x, | y |, | A |)</math>
- for every <math>(x, y, A)</math>, and a fixed function <math>a(x, |y|, |A|)</math> increasing in <math>|y|</math> and <math>|A|</math>, and locally integrable in <math>x</math>. If <math>J</math> is sequentially weakly lower semi-continuous, then for any given <math>(x, y) \in \Omega \times \mathbb{R}^m</math> the function <math>A \mapsto F(x, y, A)</math> is quasiconvex. The claim is true even when both <math>m, n</math> are bigger than <math>1</math> and coincides with the previous claim when <math>m = 1</math> or <math>n = 1</math>, since then quasiconvexity is equivalent to convexity.
Notes
References and further reading
- Шаблон:Cite book
- Шаблон:Cite book
- Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin Шаблон:ISBN.
- Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, Шаблон:ISBN.
- Шаблон:Cite news
- Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145
