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arXiv · 2310.20610

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

Ondřej Bouchala·Stanislav Hencl·Zheng Zhu
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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 $r\in \left(\tfrac5{16},\tfrac{7}{16}\right)$ the degree of $h$ with respect to the ball $B(0,r)$ is equal to $k$ on a set of positive measure.

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Ondřej Bouchala, Stanislav Hencl, Zheng Zhu. 2023-10-31. Weak Limit of $W^{1,2}$ Homeomorphisms in $\mathbb{R}^3$ Can Have Any Degree. https://arxiv.org/abs/2310.20610

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