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New lower semicontinuity results for polyconvex integrals. (English) Zbl 0810.49014
The paper deals with integral functionals of the form \(F(u,\Omega)= \int_ \Omega f(\nabla u)dx\) initially defined for \(u\in C^ 1(\Omega; \mathbb{R}^ k)\), \(\Omega\subseteq \mathbb{R}^ n\). The function \(f\) is assumed to be polyconvex in the sense of Ball, i.e., there exists a convex function \(g\) defined in the space of all \(n\)-vectors of \(\mathbb{R}^ n\times \mathbb{R}^ k\), such that \(f(A)= g({\mathcal M}(A))\) for every \(k\times n\) matrix \(A\), where \({\mathcal M}(A)\) is the \(n\)-vector whose components are the determinants of all minors of the matrix \(A\). The aim of the paper is to study the \(L^ 1(\Omega; \mathbb{R}^ k)\)-lower semicontinuous extension \(\mathcal F\) for \(F\) defined as the greatest lower semicontinuous functional on \(L^ 1(\Omega; \mathbb{R}^ k)\) which is less or equal to \(F\), where \(F(u,\Omega) \triangleq +\infty\;\forall u\in L^ 1(\Omega; \mathbb{R}^ k)\backslash C^ 1(\Omega; \mathbb{R}^ k)\). It is shown that, if the polyconvex function \(f\) satisfies the inequality \[ c_ 0| {\mathcal M}(A)|\leq f(A)\leq c_ 1(|{\mathcal M}(A)|+1) \] with \(0< c_ 0\leq c_ 1\), then for every function \(u\in \text{BV}(\Omega; \mathbb{R}^ k)\) with \(\int_ \Omega| {\mathcal M}(\nabla u)| dx< \infty\) the integral representation \[ {\mathcal F}(u,\Omega)= \int_ \Omega f(\nabla u) dx \] holds. At the same time an example is given which shows that when \(f(A)= | {\mathcal M}(A)|\) and \(k\geq 2\), the set function \(\Omega\to {\mathcal F}(u,\Omega)\) is not subadditive if \(u\) is an arbitrary function of \(\text{BV}_{\text{loc}}(\mathbb{R}^ n; \mathbb{R}^ k)\) so that the functional \(u\to {\mathcal F}(u,\Omega)\) cannot be represented by an integral over \(\Omega\). Moreover, a counterexample to the subadditivity of \(\Omega\to {\mathcal F}(u,\Omega)\) is constructed where \(u\in W^{1,p}_{\text{loc}}(\mathbb{R}^ n; \mathbb{R}^ k)\) and \(p< \min\{n,k\}\).

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
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