Weak lower semicontinuity of polyconvex integrals: A borderline case
Let \(g\) be a convex function satisfying the inequality \(g(t)\geq a| t|- b\) for suitable constants \(a> 0\) and \(b\geq 0\). We prove that, on the space \(W^{1,n}(\Omega; \mathbb{R}^ n)\), the functional \(\int_ \Omega g(\text{det } Du) dx\) is lower semicontinuous along sequences converging in \(L^ 1(\Omega; \mathbb{R}^ n)\) and bounded in \(W^{1, p}(\Omega; \mathbb{R}^ n)\), with \(p\geq n- 1\). If \(p> n- 1\), the result was proved by \textit{B. Dacorogna} and \textit{P. Marcellini} [C. R. Acad. Sci., Paris, Ser. I 311, No. 6, 393-396 (1990; Zbl 0723.49007)]. If \(p< n- 1\), the result is false, as recently shown by \textit{J. Malý} [Proc. R. Soc. Edinb., Sect. A 123, No. 4, 681-691 (1993; Zbl 0813.49017)].
scientific article; zbMATH DE number 784093