Weak limit of W^1,2 homeomorphisms in R^3 can have any degree

Abstract

In this paper for every \(k\in\mathbb{Z}\) we construct a sequence of weakly converging homeomorphisms \(h_m\colon B(0,10)\to\mathbb{R}^3\), \(h_m\rightharpoonup h\) in \(W^{1,2}(B(0,10))\), such that \(h_m(x)=x\) on \(\partial B(0,10)\) and for every \(r\in(5/16,7/16)\) the degree of \(h\) with respect to the ball \(B(0,r)\) is equal to \(k\) on a set of positive measure.