A direct proof for lower semicontinuity of polyconvex functionals
The authors discuss the lower semicontinuity of the functional \(\int_\Omega f(x,u(x),Du(x))dx\), where \(\Omega\subset\mathbb{R}^n\) is open bounded and \(f:\Omega\times\mathbb{R}^N\times \mathbb{R}^{n\times N}\to[0,+\infty)\) is polyconvex in the last variable. The main result (under suitable assumptions on \(f(\cdot,\cdot,A)\) is the lower semicontinuity with respect to a sequence \((u_h)_{h\in\mathbb{N}}\) in \(W^{1,k}(\Omega;\mathbb{R}^N)\), \(k=\min\{n,N\}\), which \(L_1\)-converges to \(u\in W^{1,1}(\Omega;\mathbb{R}^N)\) and such that \(\sup_{h\in\mathbb{N}}\int_\Omega|{\mathcal M}(Du_h)|dx<+\infty\), where \(\mathcal M\) denotes the vector of the minors of gradient matrix. As a particular case, the authors prove the lower semicontinuity of the integral \(\int_\Omega g(\text{det} Du(x))dx\) in the case \(g:\Omega\to [0,+\infty)\) is convex and satisfies \(g(v)\geq a|v|-b\). The authors observe that results on the same subject had already been proved by \textit{G. Dal Maso} and \textit{C. Sbordone} [Math. Z. 218, No. 4, 603-609 (1995; Zbl 0822.49010)]. The main advantage of the paper under review is a different approach which makes use of blow-up arguments instead of the theory of Cartesian currents and allows a direct and simpler proof.
scientific article; zbMATH DE number 784060
scientific article