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We state necessary and sufficient conditions for weak lower semicontinuity of$u\mapsto\int_\Omega h(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ iscontinuous and possesses a recession function, and $u\inL^p(\Omega;\mathbb{R}^m)$, $p>1$, lives in the kernel of a constant-rankfirst-order differential operator $\mathcal{A}$ which admits an extensionproperty. Our newly defined notion coincides for$\mathcal{A}=\operatorname{curl}$ with quasiconvexity at the boundary due toJ.M. Ball and J. Marsden. Moreover, we give an equivalent condition for weaklower semicontinuity of the above functional along sequences weakly convergingin $L^p(\Omega;\mathbb{R}^m)$ and approaching the kernel of $\mathcal{A}$ evenif $\mathcal{A}$ does not have the extension property.

$\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals