(i) Let be a bounded region in with smooth boundary . Show that Poisson's equation in
has at most one solution satisfying on , where and are given functions.
Consider the alternative boundary condition on , for some given function , where is the outward pointing normal on . Derive a necessary condition in terms of and for a solution of Poisson's equation to exist. Is such a solution unique?
(ii) Find the most general spherically symmetric function satisfying
in the region for . Hence in each of the following cases find all possible solutions satisfying the given boundary condition at : (a) , (b) .
Compare these with your results in part (i).