Paper 3, Section II, C

Vector Calculus | Part IA, 2012

(i) Let VV be a bounded region in R3\mathbb{R}^{3} with smooth boundary S=VS=\partial V. Show that Poisson's equation in VV

2u=ρ\nabla^{2} u=\rho

has at most one solution satisfying u=fu=f on SS, where ρ\rho and ff are given functions.

Consider the alternative boundary condition u/n=g\partial u / \partial n=g on SS, for some given function gg, where nn is the outward pointing normal on SS. Derive a necessary condition in terms of ρ\rho and gg for a solution uu of Poisson's equation to exist. Is such a solution unique?

(ii) Find the most general spherically symmetric function u(r)u(r) satisfying

2u=1\nabla^{2} u=1

in the region r=rar=|\mathbf{r}| \leqslant a for a>0a>0. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at r=ar=a : (a) u=0u=0, (b) un=0\frac{\partial u}{\partial n}=0.

Compare these with your results in part (i).

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