Homework 5A: Green’s functionsMath 456/556

1. (The vortex patch) The so-called streamfunction ψ : R2 → R in fluidmechanics solves ∆ψ = ω where ω is a measure of the fluid rotation (thevelocity field is perpendicular to the gradient of ψ, by the way). A simplemodel of a hurricane of radius R has

ω =

{1 |x| < R,

0 |x| ≥ R..

Use the Green’s function appropriate for two dimensions to express thesolution ψ as an integral in polar coordinates. You will want to use the lawof cosines

|x− x0| = |x0|2 + |x|2 − 2|x||x0| cos(ϕ)

where ϕ is the angle between vectors x and x0. Finally, evaluate ψ(0) (youcan actually evaluate the integral in general, but its a lot more compli-cated!).

2. (The Helmholtz equation in 3D) We want the Green’s function G(x;x0)for the equation ∆u− k2u = f(x) in R3, where k is a constant.(a) Like the Laplace equation, we suppose that G only depends on the dis-tance from x to x0, that is G(x;x0) = g(r) where r = |x − x0|. Going tospherical coordinates, we find g satisfies the ordinary differential equation

1

r2(r2gr)r − k2g = 0, r > 0.

The trick to solving this is the change of variables g(r) = h(r)/r. The equa-tion for h(r) has simple exponential solutions.(b) The constant of integration in part (a) is found just like the LaplacianGreen’s function in R3, using the normalization condition

limr→0

∫Sr(0)

∇xG(x; 0) · n̂ dx = 1,

where Sr(0) is a spherical surface of radius r centered at the origin. (Recallthis comes from integrating the equation for G on the interior of Sr(0) andapplying the divergence theorem).(c) Finally, write down the solution to ∆u − k2u = f(x) assuming u → 0as |x| → ∞. Write the answer explicitly as iterated integrals in Cartesiancoordinates.

3. Using the method of images, the Green’s function for the Laplacian on asphere |x| < a for problems with Dirichlet boundary conditions is

G(x,x0) = − 1

4π|x− x0|+

(a/r0)

4π|x− a2x0/r20|, r0 = |x0|.

In the case x0 = 0, the correct formula is G(x, 0) = −1/(4π|x|) + 1/(4πa)(a) Check that G is indeed correct: show formally that ∆G = δ(x − x0), atleast restricted to the domain |x| < a. Hint: recall

∆

(1

4π|x− x0|

)= −δ(x− x0).

(b) show G(x;x0) = 0 if |x| = a, i.e. the Green’s function satisfies theDirichlet boundary condition. Hint: rewrite each norm using the law ofcosines, and then set |x| = a.(c) It turns out that the solution to ∆u = 0 can be written in terms of theboundary values of u by the formula

u(x0) =

∫|x|=a

u(x)∇xG(x;x0) · n̂ dx.

Use this to show that that u(0) is just the average of the values of u on thespherical surface |x| = a. This just generalizes the so-called mean valuetheorem for the two-dimensional Laplace equation.

4. Find the Green’s function for ∆u = f(x, y) in the first quadrant x >0, y > 0 where the boundary/far-field conditions are

u(x, 0) = 0, u(0, y) = 0, lim|x|→∞

|∇u| = 0.

(Hint: locate image sources in each quadrant, choosing their signs appro-priately)