We state necessary and su cient conditions for weak lower semicontinuity of integral functionals of the form u → ∫ Ω h (x, u(x)) dx, where h is continuous and possesses a positively p-homogeneous recession function, p > , and u ∈ L p (Ω; ℝ m ) lives in the kernel of a constant-rank rst-order di erential operator A which admits an extension property. In the special case A = curl, apart from the quasiconvexity of the integrand, the recession function's quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly de ned notions of A-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in L p (Ω; ℝ m ) and approaching the kernel of A even if A does not have the extension property.

Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappingsMoreover, the constraint det Y k > 0 can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if Y := ∇y for y ∈ W 1,p (Ω; R n ). Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in W 1,∞ (Ω; R n ) where the inverted gradients are also bounded in L ∞ Ω; R n×n ). This extends the original results due to D. Kinderlehrer and P. Pedregal [19].

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