Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

Abstract

In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds, $n\geq 2$, under the assumption that the Young function $P$ satisfies the so called divergence condition $\int_1^\infty P(t)/t^{n+1}\, dt=\infty$. We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\in L^n$ and, more generally, mappings with $Df\in L^n\log^{-1}L$. On the other hand, if the space $W^{1,P}$ is larger than $W^{1,n}$ (for example if $Df\in L^n\log^{-1}L$), and the universal cover of $N$ is homeomorphic to $\mathbb{S}^n$, $n\neq 4$, or is diffeomorphic to $\mathbb{S}^n$, $n=4$, then we construct an example of a mapping in $W^{1,P}(M,N)$ that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold $N$.