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. 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. We let { f k } k = 1 ∞ {displaystyle {f_{k}}_{k=1}^{infty }} be a bounded sequence in L ∞ ( U , R m ) {displaystyle L^{infty }(U,mathbb {R} ^{m})} , where U {displaystyle U} denotes an open bounded subset of R n {displaystyle mathbb {R} ^{n}} . Then there exists a subsequence { f k j } j = 1 ∞ ⊂ { f k } k = 1 ∞ {displaystyle {f_{k_{j}}}_{j=1}^{infty }subset {f_{k}}_{k=1}^{infty }} and for almost every x ∈ U {displaystyle xin U} a Borel probability measure ν x {displaystyle u _{x}} on R m {displaystyle mathbb {R} ^{m}} such that for each F ∈ C ( R m ) {displaystyle Fin C(mathbb {R} ^{m})} we have F ∘ f k j ⇀ ∗ ∫ R m F ( y ) d ν ⋅ ( y ) {displaystyle Fcirc f_{k_{j}}{overset {ast }{ ightharpoonup }}int _{mathbb {R} ^{m}}F(y)d u _{cdot }(y)} in L ∞ ( U ) {displaystyle L^{infty }(U)} . The measures ν x {displaystyle u _{x}} are called the Young measures generated by the sequence { f k } k = 1 ∞ {displaystyle {f_{k}}_{k=1}^{infty }} . For every minimizing sequence u n {displaystyle u_{n}} of I ( u ) = ∫ 0 1 ( u x 2 − 1 ) 2 + u 2 d x {displaystyle I(u)=int _{0}^{1}(u_{x}^{2}-1)^{2}+u^{2}dx} subject to u ( 0 ) = u ( 1 ) = 0 {displaystyle u(0)=u(1)=0} , the sequence of derivatives u n ′ {displaystyle u'_{n}} generates the Young measures ν x = 1 2 δ − 1 + 1 2 δ 1 {displaystyle u _{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 ± 1 {displaystyle pm 1} (or close to ± 1 {displaystyle pm 1} ).