On non-homogeneous Robin reflection for harmonic functions

Abstract

This paper concerns the reflection of harmonic functions, , defined in a neighborhood of a real-analytic curve in the plane subject to the Robin condition, , on that curve. Here a and b are constants, and is the restriction of a holomorphic function onto the curve. For the case, when , 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 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 . 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.