A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫Ωg(detDu)dx with respect to sequences of functions fromW1,n (Ω;ℝn) which converge inL1 (Ωℝn) and are uniformly bounded inW1,n−1 (Ω;ℝn), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results.

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