## Weak lower semicontinuity of polyconvex integrals: A borderline case.(English)Zbl 0822.49010

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 B. Dacorogna and 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 J. Malý [Proc. R. Soc. Edinb., Sect. A 123, No. 4, 681-691 (1993; Zbl 0813.49017)].

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

lower semicontinuity; relaxation

### Citations:

Zbl 0723.49007; Zbl 0813.49017
Full Text:

### References:

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