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. Young measures have applications in the calculus of variations 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, however, in terms of linear functionals already in 1937 still before the measure theory has been developed.

Definition

We let ${\displaystyle \{f_{k}\}_{k=1}^{\infty }}$ be a bounded sequence in ${\displaystyle L^{\infty }(U,\mathbb {R} ^{m})}$, where $U$ denotes an open bounded subset of $\mathbb {R} ^{n}$. Then there exists a subsequence ${\displaystyle \{f_{k_{j}}\}_{j=1}^{\infty }\subset \{f_{k}\}_{k=1}^{\infty }}$ and for almost every $x\in U$ a Borel probability measure ${\displaystyle \nu _{x}}$ on $\mathbb {R} ^{m}$ such that for each ${\displaystyle F\in C(\mathbb {R} ^{m})}$ we have ${\displaystyle F(f_{k_{j}}){\overset {\ast }{\rightharpoonup }}\int _{\mathbb {R} ^{m}}F(y)d\nu _{\cdot }(y)}$ in ${\displaystyle L^{\infty }(U)}$. The measures ${\displaystyle \nu _{x}}$ are called the Young measures generated by the sequence ${\displaystyle \{f_{k}\}_{k=1}^{\infty }}$.

Example

For every minimizing sequence $u_{n}$ of ${\displaystyle I(u)=\int _{0}^{1}(u_{x}^{2}-1)^{2}+u^{2}dx}$ subject to $u(0)=u(1)=0$ , 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 $\pm 1$ (or close to $\pm 1$).

References

• Ball, J. M. (1989). "A version of the fundamental theorem for Young measures". In Rascle, M.; Serre, D.; Slemrod, M. 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.
• 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 (1997). Relaxation in Optimization Theory and Variational Calculus. 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", Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, XXX (7–9): 211–103, JFM 63.1064.01, Zbl 0019.21901, 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 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

• Hazewinkel, Michiel, ed. (2001), "Young measure", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4