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In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, 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 In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
, as well as in various
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
(or optimal control problems). They are named after
Laurence Chisholm Young Laurence Chisholm Young (14 July 1905 – 24 December 2000) was a British mathematician known for his contributions to measure theory, the calculus of variations, optimal control theory, and potential theory. He was the son of William Henry Yo ...
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 Uis 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

* * * * * * * *, memoir presented by Stanisław Saks at the session of 16 December 1937 of the
Warsaw Society of Sciences and Letters Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is official ...
. The free PDF copy is made available by th
RCIN –Digital Repository of the Scientifics Institutes
*.


External links

* {{Measure theory Measures (measure theory)