On mappings in the Orlicz--Sobolev classes on Riemannian manifolds
The authors' purpose in this paper is to prove the extendability of lower \(Q\)-homeomorphisms of domains on smooth Riemannian manifolds. They begin by establishing a criterion for a homeomorphism to satisfy their lower \(Q\)-condition: Theorem (1, Thm. 3.1). Let \(D\) and \(D_*\) be domains on smooth Riemannian manifolds \((\mathbb M,g)\) and \((\mathbb{M}^n_*, g^*)\), \(n\geq 3\), respectively, and let \(\varphi: [0,\infty)\to [0,\infty)\) be an increasing function under condition (2.1). Then every homeomorphism \(f : D\to D_*\) with finite distortion of the class \(W^{1,\varphi}_{\text{loc}}(D)\) is a lower \(Q\)-homeomorphism in \(D\) with \(Q(x)= K^f(X)\). Here \(W^{1,\varphi}_{\text{loc}}(D)\) refers to the Orlicz-Sobolev class associated with the growth function \(\varphi\) and domain \(D\). The authors proceed in Section 6 to show that under the assumption of finite mean oscillation, homeomorphisms in a given Orlicz-Sobolev class extend. In the final section, finite mean oscillation is replaced by finite distortion, with similar conclusions. The flavor of these extendability theorems is represented by Theorem (1, Thm. 7.1). Let \(D\) be locally connected on the boundary, let \(\overline D\) be compact, and let \(\partial D_*\) be weakly flat. If \(f: D\to D_*\) is a homeomorphism with finite distortion of the Orlicz-Sobolev class \(W^{1,\varphi}_{\text{loc}}\) under condition (2.1), and \(K(x,f)\in L^{n-1}(D)\), then \(f^{-1}\) has a continuous extension to \(\overline D\).
scientific article; zbMATH DE number 6110536
scientific article