Convex integrands of div-curl couples (Q2704047)

The authors establish some result related to Iwaniec's conjecture on the sharp constant in the \(n\)-dimensional version of the Hilbert transform associated with the Hodge decomposition of the elliptic complex NEWLINE\[NEWLINE{\mathcal D}'({\mathbf R}^n, {\mathbf R}) @>\Delta>> {\mathcal D}' ({\mathbf R}^n, {\mathbf R}^n) @>\text{curl}>> {\mathcal D}'({\mathbf R}^n, {\mathbf R}^{n \times n}).NEWLINE\]NEWLINE The obtained result is that every quasiconvex function is convex in singular directions, where a real valued function \(f\) is said to be quasiconvex if for every constant vectors \(A\), \(B\in {\mathbf R}^n\) NEWLINE\[NEWLINE\int_{{\mathbf R}^n} \biggl[f \bigl(A+\nabla \alpha,B+ (\text{curl})^* \beta\bigr)-f(A,B) \biggr] dx\geq 0NEWLINE\]NEWLINE whenever \(\alpha\) and \(\beta\) are functions with compact support in \(W^{1,\infty} ({\mathbf R}^n, {\mathbf R})\) and \(W^{1,\infty}({\mathbf R}^n, {\mathbf R}^{n\times n})\) respectively, and \(f\) is said to be convex in singular directions if \(f(A+tX, B+tY)\) is a convex function of the real variable \(t\) whenever \(A,B,X,Y\in {\mathbf R}^n\) and \(X\) is orthogonal to \(Y\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].