Young measure

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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[]

We let ${\displaystyle \{f_{k}\}_{k=1}^{\infty }}$ be a bounded sequence in ${\displaystyle L^{p}(U,\mathbb {R} ^{m})}$, where ${\displaystyle U}$ denotes an open bounded subset of ${\displaystyle \mathbb {R} ^{n}}$. 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 weak limit exists (or weakly star 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 ${\displaystyle \{f_{k_{j}}\}_{j=1}^{\infty }}$. More generally for any Caratheodory function ${\displaystyle f(x,A):U\times R^{m}\to R}$ the limit of ${\displaystyle \int _{U}f(x,f_{j}(x))\ dx}$ if it exists will be given by ${\displaystyle \int _{U}\int _{\mathbb {R} ^{m}}f(x,A)\ d\nu _{x}(A)\ dx}$ ^[2]

Young's original idea in the case where ${\displaystyle f}$ is in ${\displaystyle C_{0}(U\times \mathbb {R} ^{m})}$ was to consider the graph of the functions and consider the uniform measure concentrated on this surface lets say ${\displaystyle \Gamma _{j}:=(id,f_{j})_{\sharp }L^{d}\llcorner U}$ , (here ${\displaystyle L^{d}\llcorner U}$is the restriction of the Lebesgue measure on ${\displaystyle U}$ )and by taking the weak star limit of these measures as elements of ${\displaystyle C_{0}(U\times \mathbb {R} ^{m})^{\star }}$ we will have ${\displaystyle \langle \Gamma _{j},f\rangle =\int _{U}f(x,f_{j}(x))\ dx\to \langle \Gamma ,f\rangle }$ 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 ${\displaystyle \nu _{x}}$.

Example[]

For every minimizing sequence ${\displaystyle u_{n}}$ of ${\displaystyle I(u)=\int _{0}^{1}(u_{x}^{2}-1)^{2}+u^{2}dx}$ subject to ${\displaystyle u(0)=u(1)=0}$ , the sequence of derivatives ${\displaystyle u'_{n}}$ generates the Young measures ${\displaystyle \nu _{x}={\frac {1}{2}}\delta _{-1}+{\frac {1}{2}}\delta _{1}}$. This captures the essential features of all minimizing sequences to this problem, namely developing finer and finer slopes of ${\displaystyle \pm 1}$ (or close to ${\displaystyle \pm 1}$).

References[]

• Ball, J. M. (1989). "A version of the fundamental theorem for Young measures". In Rascle, M.; Serre, D.; Slemrod, M. (eds.). PDEs and Continuum Models of Phase Transition. Lecture Notes in Physics. 344. Berlin: Springer. pp. 207–215.
• C.Castaing, P.Raynaud de Fitte, M.Valadier (2004). Young measures on topological spaces. Dordrecht: Kluwer.CS1 maint: multiple names: authors list (link)
• L.C. Evans (1990). Weak convergence methods for nonlinear partial differential equations. Regional conference series in mathematics. American Mathematical Society.
• S. Müller (1999). Variational models for microstructure and phase transitions. Lecture Notes in Mathematics. Springer.
• P. Pedregal (1997). Parametrized Measures and Variational Principles. Basel: Birkhäuser. ISBN 978-3-0348-9815-7.
• T. Roubíček (2020). Relaxation in Optimization Theory and Variational Calculus (2nd ed.). Berlin: W. de Gruyter. ISBN 3-11-014542-1.

• Valadier, M. (1990). "Young measures". Methods of Nonconvex Analysis. Lecture Notes in Mathematics. 1446. Berlin: Springer. pp. 152–188.
• Young, L. C. (1937), "Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations", , Classe III, XXX (7–9): 211–234, JFM 63.1064.01, Zbl 0019.21901, memoir presented by Stanisław Saks at the session of 16 December 1937 of the . The free PDF copy is made available by the RCIN –Digital Repository of the Scientifics Institutes.
• Young, L. C. (1969), Lectures on the Calculus of Variations and Optimal Control, Philadelphia–London–Toronto: W. B. Saunders, pp. xi+331, MR 0259704, Zbl 0177.37801.

External links[]

• "Young measure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]