Math 173:
Homework 2 (Due: April 24th)
Solve 5 problems for full credit. Do 7 for extra.
1. Prove that if |u(x)| ≤ C(|x|m + 1) for all x ∈ Rn , where m ∈ N, and u is harmonic in Rn ,
then u is a polynomial of degree less than or equal to m.
2. Let Ω ⊂ Rn be a bounded open set. Consider a sequence un of harmonic functions such that
un → u uniformly on U . Prove that u is an harmonic function.
3. Let u : Rn → R be a harmonic function. Prove that
Z
u2 (x) dx < +∞, then u ≡ 0. (Hint: Note that u2 must be subharmonic and use
(a) If
Rn
problem 5 from Homework 1).
Z
(b) If
|∇u2 (x)| dx < +∞, then u is constant.
Rn
4. Suppose Ω is a bounded domain in Rn and consider the operator:
X
X
Lv =
aij (x)∂i ∂j v +
bi (x)∂i v
i,j
i
where aij and bi are continuous functions on Ω and the matrix (aij )(x) is positive definite for
each x ∈ Ω.
(a) Suppose v is a function in C 2 (Ω) satisfying Lv > 0. Prove that v cannot have a local
maximum in Ω. (Hint: Use a change of coordinates to make aij diagonal at a point if
necessary.)
(b) Show that if x0 ∈
/ Ω and M > 0 is a sufficiently large constant, then w(x) = exp(−M |x−
x0 |2 ) satisfies Lw > 0 in Ω.
(c) Use (a) and (b) to prove: Suppose u ∈ C 2 (Ω) ∩ C(Ω) and Lu = 0 in Ω, then
max u = max u.
∂Ω
Ω
5. Solve problem 9 of Evans, page 87 (1st edition).
6. Suppose u is a continuous function on Ω. If for each x ∈ Ω there is a sequence of positive
numbers rj → 0 such that
Z
u(x) = − u(y) dS(y),
∂Brj (x)
for all j, then u is harmonic on Ω. (Hint: try to use the fact that we can solve the Dirichlet
problem in a ball.)
1
7. Let u be a harmonic function in Ω\{x0 } and assume that u = o(Φ(x − x0 )) as x → x0 , where
Φ is the fundamental solution of the Laplacian equation (with singularity at 0), that is:
lim u(x)|x − x0 |n−2 = 0, if n ≥ 3; or lim
x→x0
x→x0
u(x)
= 0, if n = 2.
log |x − x0 |
Prove that x0 is a removable singularity for u.
8. Consider an increasing sequence un : B(0, R) → R of harmonic functions, that is un (x) ≤
un+1 (x) for every x ∈ B(0, R). Assume that un (0) is a Cauchy sequence. Show that there
exists a harmonic function u such that un → u uniformly in B(0, r) for every r < R.
2