Description

Haoran Sun (haoransun@link.cuhk.edu.cn)
Problem 1 (P197 Q11).
(a) Easy to verify that (18) satisfies ∆G = 0 except at x = x0, G(x)|∂D = 0, G(x)−log|x−x0|/2π finite at x0. (b) Note that

Since n = x/|x|, we have
2
1 a − r0 cosϕ 1 cosϕ
a
Therefore we have proved Poisson’s formula since
u(x0) = a − ‹ dσ u(x) ∂ G(x,x0) = a − ‹ dσ
2πa ∂D ∂n
Problem 2 (P197 Q13). The Green’s function for the whole ball is
a 1
G(x,x −2πρ | x0| 4πρ∗
Reflect the green’s function wrt xy plane, we have
1 a 1 1 a 1
G(x,x0) = −4πρ + |x0| 4πρ∗ + 4πρz − |x0| 4πρ∗z
where z| z|
where x | | , and x0z is the reflection of x0 wrt xy plane, x0∗z is the reflection of x0∗ wrt xy plane.
Problem 3 (P337 Q1). Easy to prove the linearity. To prove the continuity, since f integrable on Ω, then ∀ϕN → ϕ, ϕN ∈ C∞(Ω) compactly supported, let F = |⟨|f|,1⟩| on Ω (since |f| also integrable), then ∀ϵ > 0, ∃N ∈ N s.t. ∀n > N, we have |ϕN(x) − ϕ(x)| < ϵ/F, and hence
|⟨f,ϕn⟩ − ⟨f,ϕ⟩| = ˆΩ dx f(x)[ϕn(x) − ϕ(x)]
ϵ
< dx |f(x)|F = ϵ
Ω
which means the map is continuous.
1 HW09 Haoran Sun

Problem 4 (P337 Q2). Linearity: direct prove by definition
⟨f′,aϕ + bψ⟩ = −⟨f,aϕ′ + bψ′⟩ = −ˆΩ dx f(x)(aϕ′(x) + bψ′(x)) = −a⟨f,ϕ′⟩ − b⟨f,ψ′⟩
= a⟨f′,ϕ⟩ + b⟨f′,ψ⟩
Continuity: since ϕN → ϕ uniformly and ϕN ∈ C∞(Ω) compactly supported, then ϕ′N → ϕ′ uniformly and ϕ′N ∈ C∞(Ω) compactly supported, hence
⟨f,ϕ′N⟩ → ⟨f,ϕ′⟩ ⇒ ⟨f′,ϕN⟩ → ⟨f′,ϕ⟩
Problem 5 (P337 Q5).
Claim. −c⟨Hx,ϕ⟩ = ⟨Ht,ϕ⟩.
Proof. Since
⟨Hx,ϕ⟩ = −¨Ω dxdt H dx ϕx
∞ x/c
⟨Ht,ϕ⟩ = −¨Ω dxdt H(x − ct)ϕt(x,t) = −ˆ0 dxˆ0 dt ϕt(x,x/c)
Hence c⟨Hx,ϕ⟩ = ⟨Ht,ϕ⟩.
Therefore
c2 ⟨Hxx,ϕ⟩ = −c2 ⟨Hx,ϕx⟩ = c⟨Ht,ϕx⟩ = −c⟨H,ϕxt⟩
⟨Htt,ϕ⟩ = −⟨Ht,ϕt⟩ = c⟨Hx,ϕt⟩ = −c⟨H,ϕtx⟩
Therefore we have ⟨Htt,ϕ⟩ = c2 ⟨Hxx,ϕ⟩, which means that H(x − ct) is a weak solution.
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