Motivation

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

${\displaystyle F\circ f_{k_{j}}(x){\rightharpoonup }\int _{\mathbb {R} ^{m}}F(y)d\nu _{x}(y)}$

weakly in ${\displaystyle L^{p}(U)}$ if the limit exists (or weakly* in ${\displaystyle L^{\infty }(U)}$ in case of ${\displaystyle p=+\infty }$). The measures ${\displaystyle \nu _{x}}$ are called the Young measures generated by the sequence ${\displaystyle \{f_{k_{j}}\}_{j=1}^{\infty }}$.

A partial converse is also true: If for each ${\displaystyle x\in U}$ we have a Borel measure ${\displaystyle \nu _{x}}$ on ${\displaystyle \mathbb {R} ^{m}}$ such that ${\displaystyle \int _{U}\int _{\mathbb {R} ^{m}}\|y\|^{p}d\nu _{x}(y)dx<+\infty }$, then there exists a sequence ${\displaystyle \{f_{k}\}_{k=1}^{\infty }\subseteq L^{p}(U,\mathbb {R} ^{m})}$, bounded in ${\displaystyle L^{p}(U,\mathbb {R} ^{m})}$, that has the same weak convergence property as above.

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

${\displaystyle \lim _{j\to \infty }\int _{U}G(x,f_{j}(x))\ dx,}$

if it exists, will be given by^[2]

${\displaystyle \int _{U}\int _{\mathbb {R} ^{m}}G(x,A)\ d\nu _{x}(A)\ dx}$.

Young's original idea in the case ${\displaystyle G\in C_{0}(U\times \mathbb {R} ^{m})}$ was to consider for each integer ${\displaystyle j\geq 1}$ the uniform measure, let's say ${\displaystyle \Gamma _{j}:=(id,f_{j})_{\sharp }L^{d}\llcorner U,}$ concentrated on graph of the function ${\displaystyle f_{j}.}$ (Here, ${\displaystyle L^{d}\llcorner U}$is the restriction of the Lebesgue measure on ${\displaystyle U.}$) By taking the weak* limit of these measures as elements of ${\displaystyle C_{0}(U\times \mathbb {R} ^{m})^{\star },}$ we have

${\displaystyle \langle \Gamma _{j},G\rangle =\int _{U}G(x,f_{j}(x))\ dx\to \langle \Gamma ,G\rangle ,}$

where ${\displaystyle \Gamma }$ is the mentioned weak limit. After a disintegration of the measure ${\displaystyle \Gamma }$ on the product space ${\displaystyle \Omega \times \mathbb {R} ^{m},}$ we get the parameterized measure ${\displaystyle \nu _{x}}$.

General definition

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

Pointwise converging sequence

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

${\displaystyle \nu _{x}=\delta _{f(x)},\quad x\in U.}$

Indeed, by dominated convergence theorem, ${\displaystyle F(f_{n}(x))}$ converges weakly* in ${\displaystyle L^{\infty }(U)}$ to

${\displaystyle F(f(x))=\int F(y)\,{\text{d}}\delta _{f(x)}}$

for any ${\displaystyle F\in C(\mathbb {R} ^{n})}$.

Sequence of sines

A less trivial example is a sequence

${\displaystyle f_{n}(x)=\sin(nx),\quad x\in (0,2\pi ).}$

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

${\displaystyle \nu _{x}(E)={\frac {1}{\pi }}\int _{E\cap [-1,1]}{\frac {1}{\sqrt {1-y^{2}}}}\,{\text{d}}y,\quad x\in (0,2\pi ),}$

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

${\displaystyle F(f_{n}){\rightharpoonup }^{*}{\frac {1}{\pi }}\int _{-1}^{1}{\frac {F(y)}{\sqrt {1-y^{2}}}}\,{\text{d}}y}$

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

Minimizing sequence

For every asymptotically minimizing sequence ${\displaystyle u_{n}}$ of

${\displaystyle I(u)=\int _{0}^{1}(u'(x)^{2}-1)^{2}+u'(x)^{2}dx}$

subject to ${\displaystyle u(0)=u(1)=0}$ (that is, the sequence satisfies ${\displaystyle \lim _{n\to +\infty }I(u_{n})=\inf _{u\in C^{1}([0,1])}I(u)}$), and perhaps after passing to a subsequence, the sequence of derivatives ${\displaystyle u'_{n}}$ generates Young measures of the form ${\displaystyle \nu _{x}=\alpha (x)\delta _{-1}+(1-\alpha )(x)\delta _{1}}$ with ${\displaystyle \alpha \colon [0,1]\to [0,1]}$ measurable. This captures the essential features of all minimizing sequences to this problem, namely, their derivatives ${\displaystyle u'_{k}(x)}$ will tend to concentrate along the minima ${\displaystyle \{-1,1\}}$ of the integrand ${\displaystyle (u'(x)^{2}-1)^{2}+u'(x)^{2}}$.