MATH 519 Homework
Fall 2013
61. Where is the mistake in the following argument? If u(x) = e−x then
u0 + u = 0 so by Fourier transformation
iyû(y) + û(y) = (1 + iy)û(y) = 0
y∈R
Since 1 + iy 6= 0 for real y, it follows that û(y) = 0 for all real y and hence
u(x) = 0.
N
62. Let h ∈ L1 (RN ) and assume that there exists L < (2π)− 2 such that
|b
h(y)| ≤ L a.e. Use the contraction mapping theorem to show that for
any g ∈ L2 (RN ) the convolution equation
Z
f (x) +
f (y)h(x − y) dy = g(x)
RN
has a unique solution f ∈ L2 (RN ).
63. Let f ∈ C(R) be 2π periodic. It then has a Fourier series in the
classical sense, but it also has a Fourier transform since f is a tempered
distribution. What is the relationship between the Fourier series and the
Fourier transform?
64. Show that E(x, t) = 21 H(t − |x|) is a fundamental solution for the
wave operator Lu = utt − uxx .
65. The fourth order operator Lu = uxxxx + 2uxxyy + uyyyy in R2 is the
biharmonic operator which arises in the theory of deformation of elastic
plates.
a) Show that L = ∆2 , i.e. Lu = ∆(∆u) where ∆ is the Laplacian.
b) Find a fundamental solution of L. (Suggestions: To solve LE = δ,
first
p solve ∆F = δ and then ∆E = F . Since F will depend on r =
x2 + y 2 only, you can look for a solution E = E(r) also.)