Math 295a Fall Term 2001
Assignment 2
1. Let C∞(−1,1) 3a≥a0>0 and y∈(−1,1). Compute G(·, y)
satisfying
−∂xa(x)∂xG(·, y)=δy
and G(−1, y) = G(1, y) = 0. Let f∈L1(−1,1) and show that u
defined through
u(x) = Z1
−1
G(x, y)f(y)dy , x ∈(−1,1)
solves (in which sense?) the following boundary value problem
(−∂xa(x)∂xu=f
u(−1) = u(1) = 0
2. Let ω∈Rand find the solution of
(∂2
x+ω2)G=δin D0(R)
[Hint: Let G=H f for f∈C∞(R) and the Heaviside function H]
3. Let T∈ E 0(Rn) with supp(T) = Kbe of finite order m∈N. Show
that
hT, φi= 0 for any φ∈ E (Rn) with ∂αφK≡0 for |α| ≤ m
4. Let Ebe 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 ∈Eand α∈K. A family of seminorms {pλ:λ∈Λ}is
called separating if
pλ(x) = 0 for each λ∈Λ implies x= 0 .
Now, a set X⊂Eis called open if for each x∈Xthere are finitely
many λj∈Λ and j>0 such that
x+∩j=1,...,m p−1
λj(0, j)⊂X .
Then the collection of all open sets defines a topology on Eand E
endowed with this topology is called locally convex space. A map
u:E→Eis then continuous iff preimages of open sets are open.
Express this in terms of the seminorms pλ. Prove that the space
E(Ω) is a locally convex space with seminorms
pK,m :K⊂Ω compact and m∈N
defined through pK,m(ϕ) = supx∈K,|α|≤mDαϕ(x).
