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, $$n\ge 2$$ , under the assumption that the Young function P satisfies the so-called divergence condition $$\int _1^\infty P(t)/t^{n+1}\, \hbox {d}t=\infty $$ . 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 $$\mathbb {S}^n$$ , $$n\ne 4$$ , 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.