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We state necessary and sufficient conditions for weak lower semicontinuity of integral functionals of the formu  ↦    ∫  Ω    h  ⁢   (  x  ,   u  ⁢   (  x  )    )   ⁢   d  x${u\mapsto\int_{\Omega}h(x,u(x))\,\mathrm{d}x}$   , where  h  is continuous and possesses a positively  p -homogeneous recession function,p  &gt;  1    ${p&gt;1}$   , andu  ∈    L  p   ⁢   (  Ω  ;   ℝ  m   )${u\in L^{p}(\Omega;\mathbb{R}^{m})}$    lives in the kernel of a constant-rank first-order differential operator   ${\mathcal{A}}$    which admits an extension property. In the special case  =  curl    ${\mathcal{A}=\mathrm{curl}}$   , 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 defined notions of   ${\mathcal{A}}$   -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 inL  p   ⁢   (  Ω  ;   ℝ  m   )${L^{p}(\Omega;\mathbb{R}^{m})}$    and approaching the kernel of   ${\mathcal{A}}$    even if   ${\mathcal{A}}$    does not have the extension property.

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