An Integral Equation Formulation of a Mixed Boundary Value Problem on a Sphere

The paper considers the boundary value problem (I): $\nabla ^2 w(\rho ,\varphi ) = 0$, $0 \leqq \rho < 1$, $0 \leqq \varphi \leqq \pi $; $w(1,\varphi ) = H_1 (\varphi ) $, $0 \leqq \varphi < \varphi _0 $; $({{\partial w} / {\partial \rho }})(1,\varphi ) = H_2 (\varphi )$, $\varphi _0 < \varphi \leqq \pi $ is on the unit sphere. A solution is sought in the form $w(\rho ,\varphi ) = ({1 / \pi })\int _0^\pi u(\rho \cos \varphi ,\rho \sin \varphi \cos \theta )d\theta $, where u satisfies $u_{xx} + u_{yy} = 0$. A Fredholm integral equation of the second kind with a weakly singular kernel is obtained for a function $g = g(\varphi )$ which determines w. Besides the derivation of the integral equation, the principal results are the following: (i) the solution w is unique, (ii) if $H_1 \in C^4 $, $H_2 \in C^2 $, and if other explicit conditions are satisfied by $H_1 $ and $H_2 $, then the character of the solution w at $(1,\varphi _0 )$ is obtained.