MA342H: Homework #2
due Wednesday, March 9
1. Let A be an open, bounded subset of R2 and suppose that u(x, y) satisfies
uxx + uyy = f (x, y) in A
u(x, y) = g(x, y) on ∂A
for some given functions f, g. Show that there exists a constant C > 0 such that
max |u| ≤ sup |g| + C sup |f |.
A∪∂A
∂A
A
Hint: Consider the function w(x, y) = a + bx2 for some suitably chosen a, b.
2. Show that the maximum principle does not hold for the wave equation utt = c2 uxx ,
where c > 0. Hint: there are separable solutions of the form u(x, t) = sin(ax) sin(bt).
3. Solve the second-order equation utt + 2uxx + 3uxt − ut − 2u = 0.
4. Let f, g, h, ϕ, ψ be some given functions and consider the equation
utt − 2uxx + uxt + ut = f (x, t),
where 0 ≤ x ≤ 1 and t ≥ 0.
Show that there is at most one solution u(x, t) which satisfies the conditions
u(0, t) = g(t),
u(1, t) = h(t),
u(x, 0) = ϕ(x),
ut (x, 0) = ψ(x).
Hint: Try to find an energy function for the case f = g = h = 0.