Math 401: Assignment 3 (Due Mon., Jan. 30 at the start of class)
1. Find the free-space Green’s function for the operator −∆ + k 2 (k > 0 a constant) in
R3 . That is, solve
(−∆ + k 2 )Gx = δx
in R3 , with Gx (y) → 0 as |y| → ∞. (Hint: look for Gx (y) in the form Gx (y) = g(r)/r,
r := |y − x|.)
2. Let Da be the disk of radius a in the plane, centred at the origin. Let Ca be its
boundary (circle of radius a).
(a) Use the method of images to find the Dirichlet (G = 0 on the boundary) Green’s
function for ∆ on Da .
(b) For a given function f (x) on Da , solve ∆u = f (x) in Da , u = 0 on Ca .
(c) For a given function g(θ) on Ca (0 ≤ θ ≤ 2π denoting the angle around Ca ),
solve ∆u = 0 in Da , u = g(θ) on Ca .
3. Let D = {(x1 , x2 ) ∈ R2 | x1 > 0, x2 > 0} be the first quadrant in the plane. Use the
method of images to solve the problem ∆u = 0 in D, ∂x∂ 2 u(x1 , 0) = f (x1 ), x1 > 0,
∂
and ∂x∂ 1 u(0, x2 ) = g(x2 ), x2 > 0 by finding the Neumann Green’s function ( ∂n
G=0
on the boundary).
4. Let G(y; x) be the (Dirichlet) Green’s function for ∆ and a region D (in R2 , say).
Prove (rigorously!) the symmetry of G: G(y; x) = G(x; y). (Hint: set v(z) := Gx (z)
and w(z) := Gy (z), and apply Green’s second identity to v and w on the region D with
1
disks of radius centred at x and y removed. Then recall that v(z) = 2π
ln |z−x|+H(z)
where H is is smooth and harmonic in D (and an analogous statement for w(z)), and
let ↓ 0.)
1