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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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References:
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