Young measure

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Short description: Measure in mathematical analysis

In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems). They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942.[1]

Definition

Motivation

The definition of Young measures is motivated by the following theorem: Let m, n be arbitrary positive integers, let [math]\displaystyle{ U }[/math] be an open bounded subset of [math]\displaystyle{ \mathbb{R}^n }[/math] and [math]\displaystyle{ \{ f_k \}_{k=1}^\infty }[/math] be a bounded sequence in [math]\displaystyle{ L^p (U,\mathbb{R}^m) }[/math][clarification needed]. Then there exists a subsequence [math]\displaystyle{ \{ f_{k_j} \}_{j=1}^\infty \subset \{ f_k \}_{k=1}^\infty }[/math] and for almost every [math]\displaystyle{ x \in U }[/math] a Borel probability measure [math]\displaystyle{ \nu_x }[/math] on [math]\displaystyle{ \mathbb{R}^m }[/math] such that for each [math]\displaystyle{ F \in C(\mathbb{R}^m) }[/math] we have

[math]\displaystyle{ F \circ f_{k_j}(x) {\rightharpoonup} \int_{\mathbb{R}^m} F(y)d\nu_x(y) }[/math]

weakly in [math]\displaystyle{ L^p(U) }[/math] if the limit exists (or weakly* in [math]\displaystyle{ L^\infty (U) }[/math] in case of [math]\displaystyle{ p=+\infty }[/math]). The measures [math]\displaystyle{ \nu_x }[/math] are called the Young measures generated by the sequence [math]\displaystyle{ \{ f_{k_j} \}_{j=1}^\infty }[/math].

A partial converse is also true: If for each [math]\displaystyle{ x\in U }[/math] we have a Borel measure [math]\displaystyle{ \nu_x }[/math] on [math]\displaystyle{ \mathbb R^m }[/math] such that [math]\displaystyle{ \int_U\int_{\R^m}\|y\|^pd\nu_x(y)dx\lt +\infty }[/math], then there exists a sequence [math]\displaystyle{ \{f_k\}_{k=1}^\infty\subseteq L^p(U,\mathbb R^m) }[/math], bounded in [math]\displaystyle{ L^p(U,\mathbb R^m) }[/math], that has the same weak convergence property as above.

More generally, for any Carathéodory function [math]\displaystyle{ G(x,A) : U\times R^m \to R }[/math], the limit

[math]\displaystyle{ \lim_{j\to \infty} \int_{U} G(x,f_j(x)) \ d x, }[/math]

if it exists, will be given by[2]

[math]\displaystyle{ \int_{U} \int_{\R^m} G(x,A) \ d \nu_x(A) \ dx }[/math].

Young's original idea in the case [math]\displaystyle{ G\in C_0(U \times \R^m) }[/math] was to consider for each integer [math]\displaystyle{ j\ge1 }[/math] the uniform measure, let's say [math]\displaystyle{ \Gamma_j:= (id ,f_j)_\sharp L ^d\llcorner U, }[/math] concentrated on graph of the function [math]\displaystyle{ f_j. }[/math] (Here, [math]\displaystyle{ L ^d\llcorner U }[/math]is the restriction of the Lebesgue measure on [math]\displaystyle{ U. }[/math]) By taking the weak* limit of these measures as elements of [math]\displaystyle{ C_0(U \times \R^m)^\star, }[/math] we have

[math]\displaystyle{ \langle\Gamma_j, G\rangle = \int_{U} G(x,f_j(x)) \ d x \to \langle\Gamma ,G\rangle, }[/math]

where [math]\displaystyle{ \Gamma }[/math] is the mentioned weak limit. After a disintegration of the measure [math]\displaystyle{ \Gamma }[/math] on the product space [math]\displaystyle{ \Omega \times \R^m, }[/math] we get the parameterized measure [math]\displaystyle{ \nu_x }[/math].

General definition

Let [math]\displaystyle{ m,n }[/math] be arbitrary positive integers, let [math]\displaystyle{ U }[/math] be an open and bounded subset of [math]\displaystyle{ \mathbb R^n }[/math], and let [math]\displaystyle{ p\geq 1 }[/math]. A Young measure (with finite p-moments) is a family of Borel probability measures [math]\displaystyle{ \{\nu_x : x\in U\} }[/math] on [math]\displaystyle{ \mathbb R^m }[/math] such that [math]\displaystyle{ \int_U\int_{\R^m} \|y\|^p d\nu_x(y)dx\lt +\infty }[/math].

Examples

Pointwise converging sequence

A trivial example of Young measure is when the sequence [math]\displaystyle{ f_n }[/math] is bounded in [math]\displaystyle{ L^\infty(U, \mathbb{R}^n ) }[/math] and converges pointwise almost everywhere in [math]\displaystyle{ U }[/math] to a function [math]\displaystyle{ f }[/math]. The Young measure is then the Dirac measure

[math]\displaystyle{ \nu_x = \delta_{f(x)}, \quad x \in U. }[/math]

Indeed, by dominated convergence theorem, [math]\displaystyle{ F(f_n(x)) }[/math] converges weakly* in [math]\displaystyle{ L^\infty (U) }[/math] to

[math]\displaystyle{ F(f(x)) = \int F(y) \, \text{d} \delta_{f(x)} }[/math]

for any [math]\displaystyle{ F \in C(\mathbb{R}^n) }[/math].

Sequence of sines

A less trivial example is a sequence

[math]\displaystyle{ f_n(x) = \sin (n x), \quad x \in (0,2\pi). }[/math]

It can be shown that the corresponding Young measure satisfies[3]

[math]\displaystyle{ \nu_x(E) = \frac{1}{\pi} \int_{E\cap [-1,1]} \frac{1}{\sqrt{1-y^2}} \, \text{d}y, \quad x \in (0,2\pi), }[/math]

for any measurable set [math]\displaystyle{ E }[/math]. In other words, for any [math]\displaystyle{ F \in C(\mathbb{R}^n) }[/math]:

[math]\displaystyle{ F(f_n) {\rightharpoonup}^* \frac{1}{\pi} \int_{-1}^1 \frac{F(y)}{\sqrt{1-y^2}} \, \text{d}y }[/math]

in [math]\displaystyle{ L^\infty((0,2\pi)) }[/math]. Here, the Young measure does not depend on [math]\displaystyle{ x }[/math] and so the weak* limit is always a constant.

Minimizing sequence

For every asymptotically minimizing sequence [math]\displaystyle{ u_n }[/math] of

[math]\displaystyle{ I(u) = \int_0^1 (u'(x)^2-1)^2 +u'(x)^2 dx }[/math]

subject to [math]\displaystyle{ u(0)=u(1)=0 }[/math] (that is, the sequence satisfies [math]\displaystyle{ \lim_{n\to+\infty} I(u_n)=\inf_{u\in C^1([0,1])}I(u) }[/math]), and perhaps after passing to a subsequence, the sequence of derivatives [math]\displaystyle{ u'_n }[/math] generates Young measures of the form [math]\displaystyle{ \nu_x= \alpha(x) \delta_{-1} + (1-\alpha)(x)\delta_1 }[/math] with [math]\displaystyle{ \alpha\colon[0,1]\to[0,1] }[/math] measurable. This captures the essential features of all minimizing sequences to this problem, namely, their derivatives [math]\displaystyle{ u'_k(x) }[/math] will tend to concentrate along the minima [math]\displaystyle{ \{-1,1\} }[/math] of the integrand [math]\displaystyle{ (u'(x)^2-1)^2 +u'(x)^2 }[/math].

References

  1. ↑ Young, L. C. (1942). "Generalized Surfaces in the Calculus of Variations". Annals of Mathematics 43 (1): 84–103. doi:10.2307/1968882. ISSN 0003-486X. https://www.jstor.org/stable/1968882. 
  2. ↑ Pedregal, Pablo (1997). Parametrized measures and variational principles. Basel: Birkhäuser Verlag. ISBN 978-3-0348-8886-8. OCLC 812613013. https://www.worldcat.org/oclc/812613013. 
  3. ↑ Dacorogna, Bernard (2006). Weak continuity and weak lower semicontinuity of non-linear functionals. Springer. 

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