MATHEMATICAL TRIPOS Part II PAPER 4 Before you begin read ...

15
30B Partial Differential Equations
i) State the Lax–Milgram lemma.
ii) Consider the boundary value problem
∆ 2 u − ∆u + u = f in Ω,
u = ∇u · γ = 0
on ∂Ω,
where Ω is a bounded domain in R n with a smooth boundary, γ is the exterior unit normal
vector to ∂Ω, and f ∈ L 2 (Ω). Show (using the Lax–Milgram lemma) that the boundary
value problem has a unique weak solution in the space
[Hint. Show that
H 2 0 (Ω) := { u : Ω → R; u = ∇u · γ = 0 on ∂Ω } .
n∑
‖∆u‖ 2 L 2 (Ω) =
∥
∂2 u
∥ ∥∥
2
∂x i ∂x j
i,j=1
L 2 (Ω)
for all u ∈ C ∞ 0 (Ω),
and then use the fact that C ∞ 0 (Ω) is dense in H2 0 (Ω).]
Part II, Paper 4
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