Conformal transformation of uniform domains under weights that depend on distance to the boundary

2022/04/05 by Ryan Gibara, Gibara, Ryan, Nageswari Shanmugalingam +1 · 1 citation
Computer Science · Engineering · #Digital Image Processing Techniques #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry

paper · pdf · doi:10.48550/arxiv.2204.01920

Abstract

The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.

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• Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces

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