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A direct proof for lower semicontinuity of polyconvex functionals. (English) Zbl 0874.49015
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 G. Dal Maso and 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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting
Zbl 0822.49010
Full Text:
##### References:
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