On non-homogeneous Robin reflection for harmonic functions

Abstract

This paper concerns the reflection of harmonic functions, $w(x,y)$, defined in a neighborhood of a real-analytic curve in the plane subject to the Robin condition, $aw+b{\mathrm{\partial }}_{n}w={\phi }_w$, on that curve. Here a and b are constants, and ${\phi }_w$ is the restriction of a holomorphic function onto the curve. For the case, when ${\phi }_w=0$, while a and b are real-analytic functions, a reflection formula was derived in Belinskiy and Savina [The Schwarz reflection principle for harmonic functions in ${\mathbb{R}}^2$ subject to the Robin condition. J Math Anal Appl. 2008;348:685–691], using the reflected fundamental solution method. Here, we construct a Robin-to-Neumann mapping and use it for obtaining the reflection operator. Since the two formulae look different, we show their equivalence when a and b are constants and ${\phi }_w=0$. As examples, we show reflection formulae for non-homogeneous Neumann and Robin conditions on the common within mathematical physics curves, such as circles and lines.

Keywords:

• Harmonic functions
• Schwarz symmetry principle
• Dirichlet-to-Neumann map
• Robin-to-Neumann map

2020 Mathematics Subject Classifications:

• 35J15
• 32D15

Acknowledgments

We are very grateful to the anonymous referee for the comments that helped improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).