Compulsory assignment, MAT-INF4300, 2007
Deadline: November 2
Everywhere below we consider the space R3 and denote by r the function r(x) = |x|.
1. Prove a removable singularity theorem: if u is harmonic on U \ {x0 } and |x − x0 |u(x) → 0 as
x → x0 , then u extends to a harmonic function on U .
For this use the Poisson integral to reduce the proof to the following statement: if u is continuous
on B(0, 1) \ {0}, harmonic in the interior, |x|u(x) → 0 as x → 0 and u|∂B(0,1) = 0 then u = 0. To
prove the latter use the maximum principle.
2. Recall that a function u on R3 \ {0} is called homogeneous of degree k ∈ R if u(λx) = λk u(x)
for all λ > 0.
Show that u ∈ C 1 (R3 \{0}) is homogeneous of degree k if and only if it satisfies the Euler equation
Su = ku, where
∂
∂
∂
S = x1
+ x2
+ x3
∂x1
∂x2
∂x3
is the scaling operator.
3. Consider the sphere S 2 ⊂ R3 . Any function f on S 2 extends uniquely to a homogeneous
˜ : C 2 (S 2 ) → C(S 2 ) by
function fk of degree k. Define a differential operator ∆
˜ = (∆f0 )|S 2 .
∆f
It is called the spherical Laplacian.
Show that for every k
˜ = (∆fk )|S 2 − (k 2 + k)f.
∆f
˜ = −(k 2 + k)f . Such functions f are called spherical
Conclude that fk is harmonic if and only if ∆f
harmonics of degree k. Note that as the equation k 2 + k = Λ has two solutions in k, the spherical
harmonics of degrees k and −1 − k coincide.
4. Use the removable singularity theorem and Liouville’s theorem to show that if k ∈ (−1, 0) then
there are no nonzero spherical harmonics of degree k.
5. Use the removable singularity theorem and analyticity of harmonic functions to show that if
f is a nonzero spherical harmonic of degree k ≥ 0 then k is an integer and fk is a homogeneous
polynomial of degree k.
6. Denote by Hk the space of homogeneous polynomials of degree k ∈ N. Show that the Laplacian
maps Hk surjectively onto Hk−2 . Conclude that the dimension of the space of spherical harmonics
of degree k ≥ 0 is 2k + 1.
7. Consider the differential operators
∂
∂
∂
∂
∂
∂
F1 = x3
− x2
, F2 = x1
− x3
, F3 = x2
− x1
.
∂x2
∂x3
∂x3
∂x1
∂x1
∂x2
(i) Show that they commute with ∆ and S. Conclude that they map the space of homogeneous
harmonic functions of degree k ∈ Z into itself.
(ii) Show that F12 + F22 + F32 = r2 ∆ − S 2 − S.
8∗ . Show that if we have a nonzero finite dimensional vector space V and operators h, e, f such
that
h2 h
he − eh = 2e, hf − f h = −2f, ef − f e = h,
+ + f e = k 2 + k (for some k ≥ 0)
4
2
then the dimension of V is at least 2k + 1. For this show that there exist an eigenvector v of h with
eigenvalue 2λ such that ev = 0, and a nonnegative integer m such that f m v 6= 0, f m+1 v = 0. Then
λ2 + λ = k 2 + k = (λ − m)2 − (λ − m), so that λ = k and m = 2k.
1
2
Denote by Vk the space of homogeneous harmonic functions of degree k (which is of dimension
2k + 1 if k ≥ 0, and of dimension −2k − 1 if k ≤ −1, by 6). Conclude that the only subspaces of
Vk invariant under the operators F1 , F2 , F3 are Vk and 0. For this consider the operators h = 2iF1 ,
e = F2 + iF3 , f = −F2 + iF3 and check that
he − eh = 2e, hf − f h = −2f, ef − f e = h, F12 + F22 + F32 = −
h2 h
− − f e.
4
2
9∗ . For k ≥ 0 consider the space Wk spanned by the functions Dα (r−1 ), |α| = k.
(i) Show that Wk ⊂ V−k−1 .
(ii) Show that Wk is invariant under F1 , F2 , F3 .
By 8 we conclude that Wk = V−k−1 . Equivalently, we see that any spherical harmonic of degree
k is obtained as a linear combination of the functions Dα (r−1 ), |α| = k, restricted to S 2 .
In fact, using the following algebraic fact: any real homogeneous polynomial of degree k can be
written in a unique way in the form f (x) + g(x)|x|2 , where f is a product of linear homogeneous
Q
polynomials (that is, it has the form kl=1 (al1 x1 + al2 x2 + al3 x3 )), one concludes the following.
Theorem (Maxwell) For any homogeneous harmonic function u of degree −k − 1 (k ≥ 0) there
exists a unique polynomial f which is a product of homogeneous linear polynomials and
∂
∂
∂
u=f
,
,
r−1 .
∂x1 ∂x2 ∂x3