Young measure

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.

Definition

We let $\_^\infty$ be a bounded sequence in $L^p (U,\mathbb^m)$, where $U$ denotes an open bounded subset of $\mathbb^n$. Then there exists a subsequence $\_^\infty \subset \_^\infty$ and for almost every $x \in U$ a Borel probability measure $\nu_x$ on $\mathbb^m$ such that for each $F \in C(\mathbb^m)$ we have $F \circ f_(x) \int_ F(y)d\nu_x(y)$ weakly in $L^p(U)$ if the weak limit exists (or weak star in $L^\infty (U)$ in case of $p=+\infty$). The measures $\nu_x$ are called the Young measures generated by the sequence $\_^\infty$. More generally, for any $f(x,A) : U\times R^m \to R,$ the limit of $\int_ f(x,f_j(x)) \ d x,$ if it exists, will be given by $\int_ \int_ f(x,A) \ d \nu_x(A) \ dx$.

Young's original idea in the case $f\in C_0(U \times \R^m)$ was to for each integer $j\ge1$ consider the uniform measure, let's say $\Gamma_j:= (id ,f_j)_\sharp L ^d\llcorner U,$ concentrated on graph of the function $f_j.$ (Here, $L ^d\llcorner U$is the restriction of the Lebesgue measure on $U.$) By taking the weak-star limit of these measures as elements of $C_0(U \times \R^m)^\star,$ we have $\langle\Gamma_j,f\rangle = \int_ f(x,f_j(x)) \ d x \to \langle\Gamma ,f\rangle,$ where $\Gamma$ is the mentioned weak limit. After a disintegration of the measure $\Gamma$ on the product space $\Omega \times \R^m,$ we get the parameterized measure $\nu_x$.

Example

For every minimizing sequence $u_n$ of $I(u) = \int_0^1 (u_x^2-1)^2 +u^2 dx$ subject to $u(0)=u(1)=0$ , the sequence of derivatives $u'_n$ generates the Young measures $\nu_x= \frac \delta_ + \frac\delta_1$. This captures the essential features of all minimizing sequences to this problem, namely .

References

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*, memoir presented by Stanisław Saks at the session of 16 December 1937 of the Warsaw Society of Sciences and Letters. The free PDF copy is made available by th
RCIN –Digital Repository of the Scientifics Institutes

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External links

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{{Measure theory

Measures (measure theory)