Math 401: Assignment 4 (Due Mon., Feb. 6 at the start of class)
1. 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
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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.)
2. Find the Dirichlet Green’s function for the square [0, π] × [0, π] (as an eigenfunction
expansion), and use it to express the solution of the boundary value problem

 ∆u = 0 0 < x1 < π, 0 < x2 < π
u=0
x1 = 0, x1 = π, x2 = π

u = g(x1 )
x2 = 0
3. Find the Dirichlet Green’s function (as an eigenfunction expansion) for the semiinfinite strip [0, 1] × [0, ∞), and use it to express the solution of the problem

∆u = 0 0 < x1 < 1, 0 < x2 < ∞



u = g(x1 )
x2 = 0
u
→
0
x

2 →∞


u=0
x1 = 0, x1 = 1
4. Consider the following Poisson-type equation with Neumann BCs in a rectangle:

 ∆u + 8u = f (x1 , x2 ) 0 < x1 < π, 0 < x2 < π
ux1 = 0
x1 = 0, π

ux2 = 0
x2 = 0, π
(a) Find the solvability condition on f .
(b) Find the modified Green’s function (in terms of an eigenfunction expansion).
(c) Express the family of solutions u in terms of the modified Green’s function.
Jan. 30
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