On the condition of \({\varLambda}\)-convexity in some problems of weak continuity and weak lower semicontinuity (Q2717740)

The paper studies the functional \(I_f(u):=\int_\Omega f(u(x)) dx\) with \(\Omega\subset\mathbb{R}^n\) an open bounded domain, \(f:\mathbb{R}^m\to\mathbb{R}\) continuous and \(u\in L^1_{\text{loc}}({\O};\mathbb{R}^m)\) such that \(P_ku:=\sum_{j=1}^m\sum_{i=1}^na_{ij}^k\partial u_j/\partial x_i=0\) for \(k=1,\dots ,N\). A special case when each \(P_k\) is of the form \(\partial u_{j(k)}/\partial v_k\) is investigated, hence every coordinate function \(u_j\) is constant along some subspace \(W_j\subset\mathbb{R}^n\). An important ``variational'' case \(P=\) curl is thus covered, too. The sequential \(L^\infty\)-weak* continuity (resp. lower semicontinuity) of \(I_f\) on the above specified space of \(u\)'s is linked with a \(\Lambda\)-affinity (resp. \(\Lambda\)-convexity) of \(f\), which means affinity (resp. convexity) of \(f\) along a collection \(\Lambda\) of lines containing the origin, i.e., along a special cone \(\Lambda\). These properties are shown equivalent to each other if and only if the subspaces \(\{W_j\}\) satisfy a certain transversality condition (resp. if \(\{W_j\}\) satisfy a certain parallelness condition). Some examples are given, too.