Required Problem Set 5: Due Date: November 3, 9:10AM
Required Problem Set 5:
Trick or Treat- Harmonic Functions Can’t Be Beat!
Due Date: November 3, 9:10AM
These problems are to test your skills at vector calculus and harmonic functions. As usual, it’s great to
work with friends, especially since several of these problems may be quite challenging. The write-up must
be in your own hand.
1. Consider the eigenvalue problem
d2 u(x)
+ (λ − x2 )u(x) = 0,
dx2
x ∈ (0, 1)
subject to the boundary conditions u0 (0) = u0 (1) = 0. Show that λ ≥ 0 and determine whether or not
λ can be zero.
2. Is it true that a harmonic function u, continuous up to and including the boundary, on a bounded,
connected, open subset Ω ⊂ R2 satisfies a minimum principle analogous to the maximum principle we
proved in class and in the extra handout? Namely, is it true that
min u ≤ min u
Ω
∂Ω
with equality in the above inequality if and only if u ≡ constant? Why or why not?
3. First, some NOTATION FOR THE PROBLEMS. We denote the ball in Rn centered at point p
and radius r as Br (p) i.e.
.
Br (p) = {x ∈ Rn ; |x − p| < r}
We denote the boundary of this ball by
.
∂Br (p) = Sr (p)
namely, the sphere of radius r around the point p. In the special case that p = (0, 0, ..., 0) and r = 1
we use the notation
.
S1 ((0, 0, ..., 0)) = S n−1
namely the n − 1 dimensional unit sphere in n-dimensional space. Finally, to avoid having to write
down things like dV and dS for volume elements, I can just use the more adult notation dµ(x) to
denote the volume element on whichever region I happen to be integrating. Sometimes people use the
notation dµk to denote the volume element on a k-dimensional surface, say, in case there is any chance
of confusion. Namely, with this notation one could write the usual divergence theorem as, say
Z
Z
n
∇ · A(x)dµ (x) =
A(x) · n(x)dµn−1 (x)
Ω
∂Ω
1
Recall form your course in vector calculus that the detailed form of what dµ looks like in any given
situation has to do with the way the hypersurface is parameterized. For this reason (and others) often
the volume element is omitted entirely so that one simply writes the divergence theorem as, say
Z
Z
∇·A=
A·n
Ω
∂Ω
You should be comfortable with any of these conventions since they are all in wide usage. Personally,
I find the last one the most elegant but it’s a largely matter of taste.
4. Show that, for Ω ⊂ Rn open, bounded and connected1 and with unit outer normal vector n, one has
Z
Z
(u∆v − v∆u)dµ =
(u∇v − v∇u) · ndµ
Ω
∂Ω
2
for all u, v ∈ C (Ω).
5. Let n > 2. Suppose that u is harmonic on Br˜(p) ⊃ Br (p). Use v = |x|2−n in the preceding identity
and Ω = { < |x| < 1} for a fixed ∈ (0, 1) to show that
Z
u(p) =
u(p + rw)dµn−1 (w)
S n−1
(Hint: Recall that on { < |x| < 1} one has n(x) =
x
|x| ).
6. Use the preceding problem, together with integration in polar coordinates to prove that, with the same
hypotheses on u as before that we have
Z
1
u(p) =
u(w)dµ(w)
vol(Br (p)) Br (p)
. R
In the above, recall that, as per usual, the volume of a set volΩ is defined to be volΩ = Ω dµ(x).
7. Actually, there’s no shortage of ways to prove the mean value property if you don’t like the idea
suggested above. Here’s another way you could try. Suppose that u ∈ C 2 (Ω) ∩ ker ∆ for Ω ⊂ Rn .
Define
Z
1
.
f (r) =
u(w)dµ(w),
p ∈ Ω, r < dist(p, ∂Ω)
volSr (p) Sr (p)
R
Show that f 0 = 0. Use this to then show that u(p) = volS1r (p) Sr (p) u(w)dµ(w).
8. A polynomial of several real variables pk (x) = p(x1 , ...., xk ) =
I ⊂ I1k for Ii ⊂ N 2 is called homogeneous of degree r if
p(λx) = λr p(x),
P
(i1 ,...,ik )⊂I
ai1 ...ik xi11 · · · xikk where
∀λ > 0
• Prove that a polynomial in k variables is homogeneous if and only if it is an eigenfunction of the
operator
k
X
∂
x·∇=
xi
∂x
i
i=1
• Prove that if p is a harmonic polynomial on Rn homogenous of degree r that the function
∆(
1 Notice
2 Before
p(x)
) = 0,
|x|2r+n−2
x ∈ Rn \ {(0, 0, ..., 0)}
simple-connectedness is not needed, a fact we exploit in the following problem.
going further I recommend you try working out a few examples to get a feeling for the notation here.
2
9. Extra Credit: This problem is NOT REQUIRED but you will get extra credit for solving it correctly.
Suppose that u ∈ C 0 (Ω)3 suppose that u has the property that, for every x ∈ Ω, there exists a sequence
of positive numbers j , with limj%∞ j = 0 such that
Z
u(x) =
u(x + j w)dµn−1 (w)
S n−1
Prove that such a u must be harmonic on Ω.
3 Notice
I did not assume that u is even differentiable, let alone twice differentiable here.
3
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