Attainable diffeomorphisms (Q1261159)

Let \(K\) be a unit cube in \(\mathbb{R}^ n\) \((n\geq 3)\) and \(\varphi:K\to K\) and orientation- and volume-preserving diffeomorphism. Then \(\varphi\) is shown to be attainable, i.e. there is a family \(\varphi_ t:K\to K\) of orientation- and volume-preserving diffeomorphisms such that \(\varphi_ 0=1\), \(\varphi_ 1=\varphi\) and \(\int^ 1_ 0\int_ K\bigl|{\partial\varphi_ t(x)\over\partial t}\bigr|^ 2dx dt\) is finite.