Further remarks on the lower semicontinuity of polyconvex integrals. (English) Zbl 0833.49013

Summary: We study some lower semicontinuity properties of polyconvex integrals of the form \[ \int_\Omega f(M(\nabla u))dx, \] where \(\Omega\subset \mathbb{R}^n\), \(u: \Omega\to \mathbb{R}^m\), and \(M(\nabla u)\) denotes the family of the determinants of all minors of the gradient matrix \(\nabla u\). In particular, we study the lower semicontinuity along sequences converging strongly in \(L^1(\Omega, \mathbb{R}^m)\) when the integrand depends only on the minors of \(\nabla u\) up to a given order, and the lower semicontinuity along sequences converging strongly in \(L^1(\Omega, \mathbb{R}^n)\) and bounded in \(W^{1, n-1}(\Omega, \mathbb{R}^n)\) in the special case \(m= n\).


49J45 Methods involving semicontinuity and convergence; relaxation
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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