Mappings of finite distortion: removable singularities (Q1424112)

A mapping \(f: G\to\mathbb R^n\), \(G\) open in \(\mathbb R^n\), of the class \(W^{1,1}_{\text{loc}}(G)\) is called a mapping of finite distortion if \(| f'(x)|^n\leq K(x) J(x,f)\). a.e., \(K(x)<\infty\) a.e. and the Jacobian \(J(x,f)\in L^1_{\text{loc}}(G)\). If \(K\in L^\infty(G)\), then \(f\) is called quasiregular. Suitable subexponential integrability conditions for \(K\) guarantee that the basic local properties of quasiregular mappings hold for mappings of finite distortion [\textit{T. Iwaniec}, \textit{P. Koskela}, \textit{G. Martin} and \textit{C. Sbordone}, J. Lond. Math. Soc. (2) 67, 123--136 (2003; Zbl 1047.30010)]. For quasiregular mappings removability results are substantially weaker than the corresponding results for analytic functions. However, sets of conformal capacity zero are removable for bounded quasiregular mappings. The authors provide a counterpart of this result for mappings of finite distortion. They balance a special capacity with the subexponential integrability of \(K\) to show that sets of zero capacity in the balanced sense are removable for this class of bounded mappings. They also show that their condition is essentially necessary by exhibiting an interesting mapping of finite distortion that has a point singularity.