On boundary extension of unclosed Orlicz-Sobolev mappings

This paper is devoted to the study of the boundary behavior of Orlicz-Sobolev classes that may not preserve the boundary under mapping. Under certain conditions, we show that these mappings have a continuous extension to the boundary of definition domain.

💡 Research Summary

The paper investigates the boundary behavior of mappings belonging to Orlicz‑Sobolev classes that are not required to preserve the boundary (i.e., “unclosed” mappings). The authors work in the setting of a bounded domain (D\subset\mathbb R^{n}) ((n\ge 2)) and a target domain (D’\subset\mathbb R^{n}). A mapping (f:D\to D’) is assumed to be open, discrete and of class (W^{1,\varphi}_{\text{loc}}(D)) with finite distortion. The central question is under what conditions such a mapping admits a continuous extension to a boundary point (b\in\partial D) (and consequently to the whole boundary).

The paper first recalls the necessary functional‑analytic background: weak derivatives, Sobolev and Orlicz‑Sobolev spaces, the inner dilation (K_{I,\alpha}(x,f)) (defined via the Jacobian and the minimal stretch), and the associated distortion function \