Links between Young measures associated to constrained sequences (Q2701815)

Let \(\omega_1\subset {\mathbb R}^N\) and \(\omega_2\subset {\mathbb R}^M\) be two domains, and let \((u_n)_n\) be a sequence of functions from \(\omega_1 \times \omega_2\) into \([\alpha,\beta]\subset {\mathbb R}\). Set \(v_n = \int_{\omega_1} u_n(x,y) dx\). Suppose that \((u_n)_n\) gives rise to the Young measure \(\nu = (\nu_{x,y})_{(x,y)\in \omega_1 \times \omega_2}\), and that \((v_n)_n\) gives rise to the Young measure \(\mu = (\mu_{y})_{(y\in \omega_2}\). Let \(f(x,y,\cdot):(0,1)\mapsto [\alpha,\beta]\) be a nondecreasing function whose distribution is \(\nu_{x,y}\), and \(g(y,\cdot):(0,1)\mapsto [\alpha,\beta]\) be a nondecreasing function whose distribution is \(\mu_{y}\). Then NEWLINE\[NEWLINE\int_0^t g(y,s) ds \leq \int_{\omega_1}\int_0^t f(x,y,s) ds dx,NEWLINE\]NEWLINE and the equality holds for \(t=1\). The author proves that this property is characteristic for Young's measures generated by sequences \((u_n)_n\) and \((v_n)_n\) defined as above.