Math 401: Assignment 5 (Due Mon., Feb. 13 at the start of class)
1. A smooth function u is called subharmonic (respectively superharmonic) if ∆u ≥ 0
(respectively ∆u ≤ 0).
(a) Show that if u is subharmonic (say, in R3 , for simplicity – though the Rn case is
essentially the same) then
u(x) ≤ the average of u over any sphere centred at x.
Hint: let A(r) be the average of u over a sphere of radius r centred at x. Compute
dA/dr by first making a change of variables like y = x + rz in the integral – then
apply the divergence theorem so that ∆u appears.
Remarks:
• a similar proof can be used to establish the mean-value property of harmonic
functions (without using Poisson’s formula)
• of course an identical statement (with signs reversed) holds for superharmonic functions
(b) Use part (a) to show that the maximum principle holds for subharmonic functions: if u is subharmonic on a (smooth, bounded, connected) region D (and
continuous on D), then if the maximum of u is attained at an interior point of
D (i.e. not on ∂D), u must be constant.
Remark: a superharmonic function obeys a corresponding “minimum principle”.
2. Let D be a smooth, open, bounded, connected region in Rn (n = 2 or 3 for concreteness). Use the maximum principle (for sub/superharmonic functions, as above)
to:
(a) establish the following ”comparison principle”: if functions u1 (x) and u2 (x) satisfy
∆u2 = f2 (x) D
∆u1 = f1 (x) D
,
,
u2 = g2 (x) ∂D
u1 = g1 (x) ∂D
f1 (x) ≤ f2 (x), and g1 (x) ≥ g2 (x), then u1 (x) ≥ u2 (x).
(b) prove that the Dirichlet Green’s function for ∆ on D is non-positive: Gx (y) ≤ 0
for y ∈ D.
3. In class, we proved the uniqueness of solutions for the (Dirichlet) Poisson problem
using the maximum principle. Here is another way to do it, and also a variant for the
Neumann problem. (As usual, D is a smooth, open, bounded, connected, region in
Rn .)
(a) Prove the uniqueness of solutions of
∆u(x) = f (x) x ∈ D
u(x) = g(x) x ∈ ∂D
by considering the difference w(x) of 2 solutions, multiplying the equation w
satisfies by w, and integrating over D.
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(b) In a similar way, prove that solutions of the Neumann problem
∆u(x) = f (x) x ∈ D
∂u
∂n (x) = g(x) x ∈ ∂D
are unique up to a constant (any 2 solutions must differ by a constant).
Feb. 6
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