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On the weak lower semicontinuity of energies with polyconvex integrands. (English) Zbl 0829.49011
Let \(f: \Omega \times \mathbb{R}^N \times \mathbb{R}^{N \times N}\to [0, \infty)\) be a Borel measurable function such that \(f(x, u, \xi)= a(x,u) g(x,\xi)\) and \(g(x, \cdot)\) is polyconvex in the last variable \(\xi\) for almost every \(x\in \Omega\). It is shown that if \(f\) is continuous, if \(a\) is bounded away from zero and if \(F(u):= \int_\Omega a(x,u) g(x, \nabla u(x)) dx\), \(u\in W^{1, N} (\Omega, \mathbb{R}^N)\), then \(F\) is weakly lower semicontinuous in \(W^{1,p}\), \(p> N-1\), in the sense that \(F(u)\leq \liminf_{n\to\infty} F(u_\nu)\) for \(u_\nu\), \(u\in W^{1, N} (\Omega, \mathbb{R}^N)\) such that \(u_\nu \rightharpoonup u\) in \(W^{1,p}\). On the contrary, if \(g\) is only a Carathéodory function then in general \(F\) is not weakly lower semicontinuous in \(W^{1,p}\) for \(N>p >N-1\). Precisely, it is shown that if \(F(u):= \int_K |\text{det} (\nabla u(x)) |dx\) where \(K\) is a compact set, then \(F\) is weakly lower semicontinuous in \(W^{1,p}\), \(N>p> N-1\), if and only if \(\text{meas} (\partial K)=0\).

49J45 Methods involving semicontinuity and convergence; relaxation