Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs

ABSTRACT

The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in ${\mathbb{C}}_{\mathrm{\infty }}$ with a particular angle $\pi /n,n\in {\mathbb{N}}^{\ast }$. With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation in these domains by applying the Green and Neumann representation formulas.

Keywords:

• Poisson equation
• harmonic Green function
• harmonic Neumann function
• parqueting-reflection principle

AMS Subject Classifications:

• 31A05
• 35A05
• 35A08
• 35J08
• 35J25

Disclosure statement

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