The Schwarz Potential of Even Dimensional Tori and Quadrature for Harmonic Functions

The modified Schwarz potential of an axially symmetric solid torus z generated by a disc ${\shadD}(a, R)$ of center " a " and radius R satisfies the equation $$ \Delta _{(\rho , w)}u + {{n-2}\over{\rho }}u_{\rho } = \chi _{{\shadD}}-{\bf T}_0, $$ where $\chi _{{\shadD}}$ represents the characteristic function of the disc ${\shadD}(a, R)$ and T 0 is a distribution supported on the ( n m 2)-dimensional sphere traced by the center of the disc. For even dimensions, a method to explicitly calculate the modified Schwarz potential, hence the corresponding distribution T 0 , is suggested, by showing that a suitable transformation of the modified Schwarz potential satisfies polyharmonic equation of order ( n m 2)/2. Quadrature formula for functions harmonic and integrable over such tori is also established and examples are provided.

Keywords:

• The Schwarz Potential
• Quadrature
• Harmonic Functions
• Polyharmonic Functions