Introduction to Partial Differential Equations
Exercise Sheet 6

Ross Ogilvie      7th October, 2024
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17.
Convoluted

The convolution of two functions f,g : n is defined by

(f g)(x) :=nf(y)g(x y)𝑑𝑦.
(a)
Let fn(x) = 0.5n for x [n1,n1] and 0 otherwise. Show that the following bounds hold
inf |y|n1g(y) (g fn)(0) sup |y|n1g(y).

(2 points)

(b)
Suppose now that g is continuous. Show that (g fn)(0) g(0) as n . (2 points)
(c)
Show that the convolution of C0-functions on n is a bilinear, commutative, and associative operation. (1+2+2 Points)
18.
Distributions
(a)
Choose any compact set K . Since it is bounded, there exists R > 0 with K [R,R]. Now choose any test function ϕ C0() with compact support in K. Since it is continuous, sup xK|ϕ(x)| is finite. Prove the following inequality (1 point)
|0ϕ(x)dx| 2Rsup xK|ϕ(x)|.
(b)
Define the Heaviside function H : by H(x) := 1 for x 0 and H(x) := 0 for x < 0. Show that the distribution associated to the Heaviside function
FH : C0() ,ϕ0ϕ(x)dx

is in fact a distribution on using part (a) and Definition 2.14 directly. (1 point)

(c)
Calculate the first and second derivatives of H as a distribution. If they are regular distributions, describe the corresponding function. (2 points)
(d)
Consider the circle C = {x2 + y2 = 1} 2. Show that
G(φ) :=Cφ𝑑𝜎

defines a distribution in 𝒟(2). Note that the 𝑑𝜎 indicates this is an integration over the submanifold C. Does there exist a locally integrable function g : 2 with

G(φ) =2gφdx

for all φ C0()? (Hint. Use Lemma 2.15) (2 Points + 2 Bonus Points)

19.
Delta Quadrant
(a)
Prove that the support of the delta distribution δ is {0}. (2 points)
(b)
Argue from Lemma 2.12 that δ is the limit of the standard mollifier (λ𝜖)𝜖>0 as 𝜖 0 as a sequence of distributions. (1 point)
(c)
Prove for any distribution F that the convolution with δ is again F. (2 points)