Semiconvex hulls of quasiconformal sets (Q2718695)

Connections between semiconvexity (polyconvexity, quasiconvexity, rank-one convexity) in the calculus of variations and the regularity of weakly quasiregular mappings have been studied by \textit{T. Iwaniec} [An approach to Cauchy-Riemann operators in \(\mathbb{R}^n\), preprint] and continued by several authors, see e.g. [\textit{K. Astala}, Doc. Math., J. DMV Extra Vol. ICM II, 617-626 (1998; Zbl 0978.30013)]. The function \(h_p(\xi)=|1-n/p||\xi|^p- |\xi|^{p- n}\text{ det }\xi\), where \(\xi\) is an \(n\times n\) matrix and \(|\xi|\) denotes the standard maximum norm of \(\xi\), as shown by T. Iwaniec to be rank-one convex for \(p\geq n/2\). T. Iwaniec conjectured \(h_p\) is quasiconvex. The author studies \(p\)-semiconvex hulls of the \(K\)-quasiconformal set \(S_K= \{\xi: |\xi|^n\leq K\text{ det }\xi\}\). Some of these are completely determined. The author also shows that \(h_p\) is not polyconvex for \(n/2\leq p< n\). There is also a discussion of the implications of the Iwaniec conjecture to the semiconvex hulls of \(S_K\).