PROBLEM SET 7, 18.155
DUE NOVEMBER 13, 2015
I defined the Sobolev spaces on the sphere – for all orders m ∈
R, by identifying them with homogoneous functions/distributions on
Rn \ {0} :
s
H s (Sn−1 ) = {ũ ∈ Hloc
(Rn \ {0}); x∂x ũ = 0}.
Still, we think of ũ as the unique homogeneous extension of u defined
on |x| = 1. To define norms on these spaces, choose 0 ≤ µ ∈ Cc∞ ((0, ∞))
such that
Z
(1)
µ2 (|x|)dx = 1
and set
kukH s = kµũkH s .
For most s this depends on µ but different choices result in equivalent
norms.
We defined the Laplacian on the sphere – the spherical Laplacian –
by
∆S u = |x|2 ∆Rn ũ, u ∈ H s (Sn−1 )
where the right side is homogeneous of degree 0.
Some hints should appear soon.
Q1 Show that the norm and inner product (u, v), on L2 (Sn−1 ) are
well-defined, independent of choice of µ.
Hint: You can do this by exchanging the order of integration.
Since µ only depends on |x| and the homogeneous terms do
not depend on |x| it is clear if you write the integral in polar
coordinates. I was prefering not to use polar coordinates, and
you an do it without by observing that the difference F of two
µ2 has integral zero and so satisfies |x|n F = (x · ∂x )f (|x|) where
f is smooth with compact support in (0, ∞). Then integrate by
parts.
Q2 Show the symmetry of ∆S , that
(∆u, v) = (u, ∆v), u, v ∈ H 2 (Sn−1 ).
1
2
PROBLEM SET 7, 18.155 DUE NOVEMBER 13, 2015
Hint: Go to the integral over Rn from the definition of the
inner product and integrate by parts. The ‘cross terms’ involving derivatives of µ2 or the factors of |x|2 make contributions
which vanish because they can be written in of x · ∂x ũ etc.
Q3 Compute the dimension of the eigenspace of ∆S with eigenvalue
k(k + n − 2).
Hint: This means count the harmonic polynomials at a given
level. Showing that the Euclidean Laplacian is surjective as a
map from homogeneous polynomials of degree k + 2 to k may
help with this. The surjectivity has an easy inductive proof if
you ‘order’ the polynomials by the number of factors of x1 say
and use another iteration (modulo terms with fewer powers of
x1 ). There are other arguments ..
Q4 Demonstrate Weyl’s law for the sphere, that the number, N (λ),
of eigenvalues (counted with multiplicity) less than λ > 0 satisfies
N (λ) = λ(n−1)/2 (c + e(λ)), e(λ) → 0 as λ → ∞.
Q5 Show that for each s = 2k, with k an integer, an equivalent
norm (and inner product) on H s (Sn−1 ) can be given in terms of
terms of the eigenvalues and eigenfunctions (spherical harmonics) of ∆S .
Q6 Guess a corresponding expression for an equivalent norm on
H s (Sn−1 ) for general s.
Q7 For what powers t > 0 is (∆S +1)−t Hilbert-Schmid on L2 (Sn−1 )?
Q8, Not required Let Vi = |x|∂xi i = 1, . . . , n, be vector fields on the sphere
and suppose that gij (ω) is symmetric, smooth on the sphere
and positive-definite as a matrix. Show that the differential
operator
X
X
(2)
Pu = −
Vi Vj +
aj Vi + b, aj , b ∈ C ∞ (Sn−1 ; R)
i,j
i
is Fredholm as a map H 2 (Sn−1 ) −→ L2 (Sn−1 ) and that for all
z ∈ C\δ, where δ ⊂ R is discrete and bounded below, (P −z)−1
is bounded on L2 (Sn−1 ).