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AARMS Course: Homework 1
Problem 1: Let Ω = Ω = {x | |x| ≤ 1} be the unit sphere in three-dimensions, and introduce the usual spherical
coordinates x ≡ (r cos φ sin θ, r sin φ sin θ, r cos θ) for 0 < r < 1 where 0 ≤ θ ≤ π is the polar angle, and 0 ≤ φ ≤ 2π.
Suppose that Ω contains four small spherically-shaped non-overlapping spherical holes Ωεj of a common radius ǫ ≪ 1
centered at some xj for j = 1, . . . , 4. Consider the following PDE where αj for j = 1, . . . , 4 are constants:
△u = 1 ,
u = 1,
u = αj ,
x ∈ Ω\ ∪4j=1 Ωεj ,
|x| = 1 .
x ∈ ∂Ωεj ,
j = 1, .., 4 .
(1) Derive a two-term asymptotic expansion for the solution to this problem in the form u = u0 + ǫu1 + · · · , when
ǫ ≪ 1. (Hint: your result will depend on the the Green’s function for the Laplacian in a sphere, which you can
calculate explicitly by the method of inages, and will involve an “unperturbed solution”, which can be readily
found.)
(2) Setting αj = 1 for j = 1, .., 4, use your asymptotic result to estimate u at x = (1/2, 0, 0), when the holes are
centered at the four symmetrically placed points x = (0, ±1/2, ±1/2) in the plane x = 0.
(3) Now suppose that each of the spherical holes is replaced by an ellipsodial shape with semi-axes ǫa, ǫb, and ǫc,
where a > 0, b > 0, and c > 0, are the same for each hole. Show how to estimate u at x = (1/2, 0, 0) now.
(Hint: Your answer here should be really brief).
(4) Finally, replace the condition u = 1 on |x| = 1 with the Neumann condition ∂r u = 0 on r = |x| = 1. Show how
to estimate u at x = (1/2, 0, 0) now.
Problem 2: In a 3-D domain, calculate a two-term asymptotic expansion for the principal eigenvalue of the
Neumann problem:
∆u + λu = 0 ,
∂n u = 0 ,
u = 0,
x ∈ Ω\Ωa ;
Z
u2 dx = 1 ,
Ω\Ωa
x ∈ ∂Ω ,
x ∈ ∂Ωa ≡ ∪N
j=1 ∂Ωεj .
Here Ωεj is an arbitrarily shaped trap of “radius” ǫ for which Ωεj → xj as ǫ → 0. The result you find should be the
same as given in Principal Result 2.1 of the Day 2 lecture notes.