On boundary non-preserving mappings with integral constraints

This manuscript is devoted to the study of mappings, satisfying the upper weighted Poletsky inequality. We study the case where the boundary of the domain may not be preserved under the mapping and, besides that, the majorant from the above inequality satisfies constraints of the integral-type. Under certain additional conditions on the definition domain and the corresponding cluster sets, we prove that families of above mappings are equicontinuous in the closure of this domain.

💡 Research Summary

The paper investigates a class of mappings that satisfy a weighted Poletsky inequality but do not necessarily preserve the boundary of their domain. While most of the existing literature on mappings with bounded or finite distortion focuses on homeomorphisms or on mappings that are open, discrete, and closed—properties that automatically guarantee boundary preservation—this work tackles the far less understood “boundary‑non‑preserving” case. The main difficulty in this setting is that the usual path‑lifting techniques, which are essential for controlling the modulus of curve families, break down when the boundary is not preserved.

To overcome this obstacle the authors introduce the notion of a ring Q‑mapping. A Lebesgue‑measurable function Q : ℝⁿ→