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Weak convergence of integrands and the Young measure representation. (English) Zbl 0757.49014
Let \(\mathbb{M}\) be the collection of the \(m\times n\) matrices and \(\varphi:\mathbb{M}\to\mathbb{R}\). The main results of this paper are as follows: Theorem 1. Let \(\varphi\) be continuous and quasi-convex and satisfy \(0\leq\varphi(A)\leq C(1+| A|^ p)\), \(A\in\mathbb{M}\), where \(1\leq p\leq\infty\). If \(u^ k\to u\) in \(H^{1,p}(\Omega)\) weakly, then \(\int_ E\varphi(\nabla u^ k)dx\leq\liminf_{k\to\infty}\int_ E\varphi(\nabla u^ k)dx\) for every measurable \(E\subset \Omega\).
Theorem 2. Let \(\varphi\) be continuous and satisfy, for some constants \(C\geq c>0\), \(\max\{c| A|^ p-1,0\}\leq\varphi(A)\leq C(1+| A|^ p)\), \(A\in\mathbb{M}\), where \(1\leq p<\infty\). Suppose that \(u^ k\to u\) in \(H^{1,p}(\Omega)\) weakly and \(\int_ \Omega\varphi(\nabla u^ k)dx=\lim_{k\to\infty}\int_ \Omega\varphi(\nabla u^ k)dx\). Let \(\nu=(\nu_ x)_{x\in\Omega}\) be a Young measure generated by \((\nabla u^ k)\). Then for any \(\psi\in\left\{\psi\in C(\mathbb{M}):\sup_ \mathbb{M}\left({|\psi(A)|\over| A|^ p+1}\right)<\infty\right\}\) the sequence \(\psi(\nabla u^ k)\) converges to \(\overline\psi\) in \(\sigma(L^ 1(\Omega)\), \(L^ \infty(\Omega))\), where \(\overline\psi(x)=\int_ \mathbb{M}\psi(A)d\nu_ x(A)\) in \(\Omega\) a.e.
Those results confirm the validity of the Young measure representation for the limit of the approximations generated by finite element methods when the energy density has appropriate polynomial growth at infinity. The question arose in the numerical analysis of equilibrium configurations of crystals with rapidly varying microstructure. Several applications are given. In particular, Young measure solutions of an evolution problem are found.
Reviewer: S.Shih (Tianjin)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35K15 Initial value problems for second-order parabolic equations
74B20 Nonlinear elasticity
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