# H\"older estimates for the Neumann problem in a domain with holes and a relation formula between the Dirichlet and Neumann problems

## TL;DR

This paper investigates how the geometry of domains with holes affects H"older regularity estimates for the Neumann problem, establishing a relation between Dirichlet and Neumann solutions in disk-like domains.

## Contribution

It provides new insights into the dependence of H"older estimates on domain geometry and links Dirichlet and Neumann problems in specific geometries.

## Key findings

- H"older regularity of solutions depends on domain geometry
- Established a relation between harmonic extensions and Neumann boundary conditions
- Analyzed problems in disk and exterior of disk geometries

## Abstract

In this paper we study the dependence of the H\"older estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the H\"older regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).