Math 295 Fall Term 2007
Assignment 2
1. It can be shown that any compactly supported distribution can be
viewed as some derivative of a continuous function. More precisely
∀u∈ E0∃f∈C(Ω) and α∈Nns.t.
hu, ϕi= (−1)|α|ZΩ
f(x)∂αϕ(x)dx ∀ϕ∈ D(Ω) ,
if supp(u)⊂Ω. Let δ∈ D(Rn) and find a simple representation of
it as some derivative of a continuous function (n≥2).
2. For u∈ D0(Rn) and v∈Rndefine τvu∈ D0(Rn) through
hτvu, ϕi=hu, τ−vϕi ∀ ϕ∈ D(Rn)
where τvϕ=ϕ(·+v). Show that τvuis well-defined and prove that
τheju−u
h−→ ∂juin D0(Rn) as h→0
for j= 1, . . . , n.
3. Let ϕ, ψ ∈ D(Ω) and α∈Nn. Determine the real numbers cαβ,
β≤αfor which the formula
∂α(ϕψ) = X
β≤α
cα,β ∂α−βϕ ∂βψ
holds. By β≤αit is meant that βj≤αjfor j= 1, . . . , n.
4. Show that u∈ D0(0,∞)given through
hu, ϕi:= X
m∈N∂mϕ(1
m), ϕ ∈ D(0,∞)
is well-defined. Show that uhas infinite order and that it cannot be
extended to a distribution of the real line.
5. Let Ω ⊂Rnbe an open and bounded domain with smooth boundary
∂Ω and denote by χΩits characteristic function. Compute ∇χΩin
the sense of distributions and determine its support.
Homework due by Wednesday, October 24 2007
