# On a Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a nonhomogeneous condition on an arc

## TL;DR

This paper develops reflection formulas for harmonic functions with nonhomogeneous boundary conditions using Dirichlet to Neumann and Robin to Neumann operators, extending classical symmetry principles.

## Contribution

It introduces a novel technique to derive reflection formulas for nonhomogeneous boundary conditions, generalizing existing symmetry principles for harmonic functions.

## Key findings

- Derived reflection formulas for nonhomogeneous Neumann conditions
- Extended Schwarz symmetry principles to nonhomogeneous Robin conditions
- Provided a new method using Dirichlet to Neumann operators

## Abstract

According to the Schwarz symmetry principle, every harmonic function vanishing on a real analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has the even continuation. There are different generalizations of the Schwarz symmetry principle. Most of them are dealing with homogeneous conditions and the Dirichlet case. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for nonhomogeneous Neumann and Robin conditions from a reflection formula subject to a nonhomogeneous Dirichlet condition.