# On H\"older continuity of mappings in domains and on boundaries

## TL;DR

This paper investigates the conditions under which certain mappings in Euclidean spaces and solutions to Beltrami equations exhibit H"older continuity, especially near boundaries, based on Dini-type conditions.

## Contribution

It establishes H"older and Lipschitz continuity for a class of spatial mappings with Dini-type characteristic and identifies boundary conditions for solutions of degenerate Beltrami equations.

## Key findings

- Mappings with Dini-type characteristics are H"older continuous in a domain.
- Generalized solutions to degenerate Beltrami equations are H"older continuous at boundary points under specific coefficient conditions.
- Conditions on complex coefficients ensure boundary regularity of solutions.

## Abstract

We study mappings with branching of a domain of Euclidean space. The H\"older and Lipschitz continuity are established for one class of spatial mappings whose characteristic satisfies the Dini type condition in a given domain. In addition, we found conditions on the complex coefficient of the degenerate Beltrami equations in the unit disk under which generalized homeomorphic solutions of this equation are H\"older continuous at the points of the boundary.