MATH 519 Homework Fall 2013

```MATH 519 Homework
Fall 2013
51. If f, g ∈ L1 (Rn ) , show that
supp (f ∗ g) ⊂ supp f + supp g
Here the sum of two sets is defined in the usual way,
A + B = {z = x + y : x ∈ A, y ∈ B}
52. Let Ω be a bounded open set and K ⊂⊂ Ω. Show that there exists
φ ∈ C0∞ (Ω) such that 0 ≤ φ(x) ≤ 1 and φ(x) ≡ 1 for x ∈ K. (Hint:
approximate the characteristic function of Σ by convolution, where Σ
satisfies K ⊂⊂ Σ ⊂⊂ Ω. Use problem #51 to prove the needed support
property.)
1
53. If Ω is a bounded open set in R3 , and u(x) = |x|
, show that u ∈
3
W 1,p (Ω) for 1 ≤ p &lt; 2 . Along the way, you should show carefully that a
∂u
distributional first derivative ∂x
agrees with the corresponding pointwise
i
derivative.
54. If u ∈ W 1,2 (0, 1), show that
|u(x) − u(y)| ≤ C
p
|x − y|
x, y ∈ [0, 1]
where C = ||u||W 1,2 (0,1) .
55. Evaluate the Fourier series
∞
X
(−1)n n sin (nx)
n=1
in D0 (R) (The result of problem 38 can be used here.) If possible, plot
some partial sums of this series.
```