Math 295a Fall Term 2001

Assignment 2

1. Let C^{∞}(−1, 1) ∋ a ≥ a_{0} > 0 and y ∈ (−1, 1). Compute G(·, y)

satisfying

−∂_{x}(a(x)∂_{x}G(·, y)) = δ_{y}

and G(−1, y) = G(1, y) = 0. Let f ∈ L^{1}(−1, 1) and show that u defined through

u(x) = ∫_{−1}^{1} G(x, y)f(y) dy , x ∈ (−1, 1)

solves (in which sense?) the following boundary value problem

\begin{cases}−∂_{x}(a(x)∂_{x}u) = f \\ u(−1) = u(1) = 0 \end{cases}

2. Let ω ∈ ℝ and find the solution of

(∂_{x}^{2} + ω^{2})G = δ in D′(ℝ)

[Hint: Let G = Hf for f ∈ C^{∞}(ℝ) and the Heaviside function H]

3. Let T ∈ E′(ℝ^{n}) with supp(T) = K be of finite order m ∈ ℕ. Show that

⟨T, ϕ⟩ = 0 for any ϕ ∈ E(ℝ^{n}) with ∂^{α}ϕ|_{K} ≡ 0 for |α| ≤ m

4. Let E be a given vector space. A function p : E → [0, ∞) is called seminorm if

p(x + y) ≤ p(x) + p(y) and p(αx) = |α| p(x)

for x, y ∈ E and α ∈ 𝕂. A family of seminorms {p_{λ} : λ ∈ Λ} is called separating if

p_{λ}(x) = 0 for each λ ∈ Λ implies x = 0 .

Now, a set X ⊂ E is called open if for each x ∈ X there are finitely many λ_{j} ∈ Λ and ϵ_{j} > 0 such that

x + ∩_{j=1,...,m} p_{λ_{j}}^{−1}((0, ϵ_{j})) ⊂ X .

Then the collection of all open sets defines a topology on E and E endowed with this topology is called locally convex space. A map u : E → E is then continuous iff preimages of open sets are open. Express this in terms of the seminorms p_{λ}. Prove that the space E(Ω) is a locally convex space with seminorms

{p_{K,m} : K ⊂ Ω compact and m ∈ ℕ}

defined through p_{K,m}(φ) = sup_{x∈K,|α|≤m} |D^{α}φ(x)| .

5. Let ψ ∈ D(ℝ^{n}) with ψ = 1 on 𝔹(0, 1) and define ψ_{k} through

ψ_{k}(x) = ψ(\frac{x}{k}) , x ∈ ℝ^{n} .

Show that ψ_{k} → 1 (k → ∞) in E(ℝ^{n}). Then prove that

D(ℝ^{n}) is dense in E(ℝ^{n}) as well as E(ℝ^{n})′ in D(ℝ^{n})′ .

The Homework is due by October 19 2001