Optimal assumptions for discreteness
The authors disprove a natural conjecture concerning the topological properties of Sobolev maps in Euclidean spaces. Consider a map \(f:\Omega\to\mathbb R^n\) of Sobolev class \(W^{1,n}\), where \(\Omega\) is a domain in \(\mathbb R^n\). If there exists a finite function \(K\) such that \(\|Df(x)\|^n\leq K(x)\det Df(x)\) for a.e. \(x\), then \(f\) is said to be a map of finite distortion. To avoid trivialities, assume that \(f\) is nonconstant. \textit{Yu. G. Reshetnyak} proved that \(f\) is discrete and open whenever \(K\in L^\infty\) [Sib. Math. J. 8 (1967), 466--487 (1968); translation from Sib. Mat. Zh. 8, 629--658 (1967; Zbl 0167.06601)]. It was widely believed that the condition could be weakened to \(K\in L^{n-1}\); see the paper under review for the history of this conjecture. Despite numerous partial results, only the case \(n=2\) was settled in the affirmative \textit{T. Iwaniec} and \textit{V. Šverák} [Proc. Am. Math. Soc. 118, No. 1, 181--188 (1993; Zbl 0784.30015)]. The authors construct, in any dimension \(n\geq 2\), a Lipschitz map \(f\) that transforms a line segment \(L\) into a point in such a way that \(K(x)\leq C d(x)^{-1} |\log d(x)|^{-1}\) where \(d(x)=\mathrm{dist}(x,L)\). The logarithmic factor ensures \(K\in L^{n-1}\) for \(n\geq 3\), thus providing a counterexample to the discreteness part of the conjecture. The question of whether \(K\in L^{n-1}\) implies openness remains open. When \(n=2\), the counterexample is given explicitly by \(f(x_1,x_2)=(|x_1|^{x_2}, \mathrm{sgn}\,(x_1) \log |x_1|)_P\) where \(|x_1|<1\), \(1<x_2<2\), and the subscript \(P\) indicates that the polar coordinates \((r,\varphi)\) are used in the codomain. Geometrically speaking, \(f\) transforms line segments orthogonal to \(L=\{x_1=0\}\) into spirals. This spiraling behavior increases the Jacobian determinant \(\det Df\) by the factor of about \(|\log d(x)|\), and thus reduces the distortion \(K\) by this logarithmic factor. The construction is more involved when \(n\geq 3\), but the basic geometric idea remains the same: the planes orthogonal to \(L\) are transformed into spiraling surfaces resembling a snail shell.
scientific article; zbMATH DE number 6148992
scientific article