Homework 5A: Green’s functions Math 456/556 1. (The vortex patch) The so-called streamfunction ψ : R2 → R in fluid mechanics solves ∆ψ = ω where ω is a measure of the fluid rotation (the velocity field is perpendicular to the gradient of ψ, by the way). A simple model of a hurricane of radius R has ( 1 |x| < R, ω= . 0 |x| ≥ R. Use the Green’s function appropriate for two dimensions to express the solution ψ as an integral in polar coordinates. You will want to use the law of 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) (you can actually evaluate the integral in general, but its a lot more complicated!). 2. (The Helmholtz equation in 3D) We want the Green’s function G(x; x0 ) for the equation ∆u − k 2 u = f (x) in R3 , where k is a constant. (a) Like the Laplace equation, we suppose that G only depends on the distance from x to x0 , that is G(x; x0 ) = g(r) where r = |x − x0 |. Going to spherical coordinates, we find g satisfies the ordinary differential equation 1 2 (r gr )r − k 2 g = 0, r2 r > 0. The trick to solving this is the change of variables g(r) = h(r)/r. The equation for h(r) has simple exponential solutions. (b) The constant of integration in part (a) is found just like the Laplacian Green’s function in R3 , using the normalization condition Z ∇x G(x; 0) · n ˆ dx = 1, lim r→0 Sr (0) where Sr (0) is a spherical surface of radius r centered at the origin. (Recall this comes from integrating the equation for G on the interior of Sr (0) and applying the divergence theorem). (c) Finally, write down the solution to ∆u − k 2 u = f (x) assuming u → 0 as |x| → ∞. Write the answer explicitly as iterated integrals in Cartesian coordinates. 3. Using the method of images, the Green’s function for the Laplacian on a sphere |x| < a for problems with Dirichlet boundary conditions is G(x, x0 ) = − 1 (a/r0 ) , + 4π|x − x0 | 4π|x − a2 x0 /r02 | 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 ), at least restricted to the domain |x| < a. Hint: recall 1 ∆ = −δ(x − x0 ). 4π|x − x0 | (b) show G(x; x0 ) = 0 if |x| = a, i.e. the Green’s function satisfies the Dirichlet boundary condition. Hint: rewrite each norm using the law of cosines, and then set |x| = a. (c) It turns out that the solution to ∆u = 0 can be written in terms of the boundary values of u by the formula Z u(x)∇x G(x; x0 ) · n ˆ dx. u(x0 ) = |x|=a Use this to show that that u(0) is just the average of the values of u on the spherical surface |x| = a. This just generalizes the so-called mean value theorem 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 |∇u| = 0. |x|→∞ (Hint: locate image sources in each quadrant, choosing their signs appropriately)