HOMEWORK 4. 516706 INTRODUCTION TO PARTIAL
DIFFERENTIAL EQUATIONS
• Due date: June 4, 2026.
• Please answer the following questions in detail, which means you need to state all
reasons you have been using.
• This homework covers Chapter 4: the Laplace equation, maximum principle, mean
value property, harmonic functions, and Green’s identities.
• Please mark your name, student ID, and question numbers clearly on your answer
sheet.
• In this homework, Ω ⊂ Rd denotes a bounded open set with piecewise C 1 boundary
unless otherwise stated.
1. (a) Let u ∈ C 2 (Ω) be a solution of
∆u = g
in Ω,
u=f
on ∂Ω.
Prove that for any other function v ∈ C 1 (Ω) so that v = f on ∂Ω, we have
ˆ
ˆ
|∇u|2
|∇v|2
+ gu dx ≤
+ gv dx.
2
2
Ω
Ω
(b) Let u ∈ C 2 (Ω) be a solution of
∆u − u3 = 0
u=f
in Ω,
on ∂Ω.
Prove that for any other function v ∈ C 1 (Ω) so that v = f on ∂Ω, we have
ˆ
ˆ
|∇u|2 u4
|∇v|2 v 4
+
dx ≤
+ dx.
2
4
2
4
Ω
Ω
2. Let Ω be a bounded open set in Rd and let u solve
∆u = −1
u=0
in Ω,
on ∂Ω.
Prove that
u≤
1
(diam Ω)2 .
2d
Here
diam Ω := sup{|x − y| : x, y ∈ Ω}
stands for the diameter of Ω.
1
2
3. Let u : Ω → R be a C 2 function so that ∆u ≥ 0 everywhere in Ω. Prove that for
any ball Br (x) ⊂ Ω, we have
ˆ
1
u(x) ≤
u(y) dS(y).
|∂Br | ∂Br (x)
4. Let u : B1 → R be harmonic and nonnegative. Prove that there exists a constant
C depending on dimension only so that for any two points x, y ∈ B1/4 ,
u(x) ≤ Cu(y).
Hint. Use the mean value property in two balls centered at x and y so that one
contains the other. This property of harmonic functions is called the Harnack
inequality.
5. Let Ω ⊂ Rd be an open set with a piecewise differentiable boundary and let u and
v be two C 2 functions in Ω such that
Verify that
u=0
on ∂Ω,
∆u = f
in Ω,
v=0
on ∂Ω,
∆v = g
in Ω.
ˆ
u(x)g(x) − v(x)f (x) dx = 0.
Ω
Hint. Use the divergence theorem.