Let S be a locally compact metric space with a measure \(\mu\). Given a function F on \(S\times {\mathbb{R}}^{\ell}\), define a functional \({\mathcal L}_ F\), \[ {\mathcal L}_ F(u)=\int_{S}F(x,u(x))d\mu (x) \] (u maps S to \({\mathbb{R}}^{\ell})\). The results have the following nature. Suppose that \(F_ m\to F_ 0\), where the \(F_ m's\) are convex and \(F_ 0\) is strictly convex in the second variable. Let \(u_ m\to u_ 0\) (in a certain weak topology) and suppose that \({\mathcal L}_{F_ m}(u_ m)\to {\mathcal L}_{f_ m}(u_ 0)\). Then \({\mathcal L}_{K_ m}(u_ m)\to {\mathcal L}_{K_ 0}(u_ 0)\) for every sequence \(\{K_ m\}\) majorized (in a certain sense) by the sequence \(\{F_ m\}\) and convergent to \(K_ 0\).