Quasiconvex functions, \(SO(n)\) and two elastic wells (Q1375389)

It is shown that the function \(F:= \text{dist}(\cdot, SO(n))^2: \mathbb{R}^{n\times n}\to \mathbb{R}\), with \(SO(n)\) denoting the special orthogonal group on \(\mathbb{R}^n\), is not quasiconvex and its quasiconvexification is bounded from below by \(\varepsilon F\) with a positive \(\varepsilon\) depending on \(n\geq 2\). A similar result holds for \(F:= \text{dist}(\cdot, SO(n)\cup SO(n)H)^2\) provided \(n\geq 3\) and \(n+ n\text{ det }H- \text{tradj }H- \text{tr }H> 0\). Stability of the two-well problem is shown in the sense that the Young measure supported on two incompatible wells remains composed from Diracs even if the wells are perturbed in a Lipschitz manner. The methods are based on \(W^{1,\infty}\)-approximation of \(W^{1,2}\)-minimizing sequences and on Šverák's idea to show the Cauchy property of the sequence.