2018/04/20 by Paweł Goldstein, Piotr Hajłasz, Goldstein, Paweł +1
Mathematics · #30C65 (46E35 #58C07) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30C65
paper · pdf · doi:10.48550/arxiv.1804.07625
arxiv created 2018/04/20 · arxiv updated 2018/04/23
In the paper we investigate continuity of Orlicz-Sobolev mappings W1,P(M,N) of finite distortion between smooth Riemannian n-manifolds, n≥ 2, under the assumption that the Young function P satisfies the so called divergence condition ∫1^∞ P(t)/tn+1 dt=∞. 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∈ Ln and, more generally, mappings with Df∈ Lnlog-1L. On the other hand, if the space W1,P is larger than W1,n (for example if Df∈ Lnlog-1L), and the universal cover of N is homeomorphic to \mathbbSn, n≠ 4, or is diffeomorphic to \mathbbSn, n=4, then we construct an example of a mapping in W1,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.