The function satisfies in and on , where is a region of which is bounded by the surface . Prove that everywhere in .
Deduce that there is at most one function satisfying in and on , where and are given functions.
Given that the function depends only on the radial coordinate , use Cartesian coordinates to show that
Find the general solution in this radial case for where is a constant.
Find solutions for a solid sphere of radius with a central cavity of radius in the following three regions:
(i) where and and bounded as ;
(ii) where and ;
(iii) where and and as .