Weak Limit of $W^{1,2}$ Homeomorphisms in ${\mathbb{R}}^3$ Can Have Any Degree

Abstract

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