Motivation edit

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 ${\displaystyle L^{p}(U,\mathbb {R} ^{m})}$ ^{[clarification needed]}. 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 edit

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 edit

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 edit

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 edit

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 ${\displaystyle (u'(x)^{2}-1)^{2}+u'(x)^{2}}$ .

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