The Neumann Problem.pdf
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The Neumann Problem
June 6, 2017
1 Formulation of the Problem
Let D be a bounded open subset in R
d
with ∂D its boundary such that D
is sufficiently nice (to be stipulated la ter as Lipschitz). Let f ∈ L
2
(D) and
g : L
2
(∂D) be two given scalar fields and n : ∂D → S
d−1
be the normal unit
vector to the boundary. Prove that
(
∆ϕ = f
(∇ϕ) · n|
∂D
= g
(1)
has a unique solution up to a constant for the unknown scalar field ϕ : D → R
in H
1
(D) if and only if
Z
D
f =
Z
∂D
g (2)
(This last condition makes sense because L
2
⊆ L
1
)
1.1 Sketch of Solution
1. Verify that if a solution of (
1) exists, then (2) must be satisfied using the
divergence theorem.
2. Formulate (1) as a variational problem: ϕ solves (1) iff
Z
D
(∇ϕ) · (∇ψ) = −
Z
D
fψ +
Z
∂D
gψ ∀ψ (3)
Assuming (
2) is satisfied.
3. Write (
3) using the bilinear and linear respectively forms
ω (ϕ, ψ) :=
Z
D
(∇ϕ) · (∇ψ)
and
η (ψ) := −
Z
D
fψ +
Z
∂D
gψ
1
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