MATH 520 Homework
Spring 2014
60. Find a functional on H 1 (Ω) for which the Euler-Lagrange equation is
−∆u = f
x∈Ω
−
∂u
= k(x)u
∂n
x ∈ ∂Ω
61. Find the Euler-Lagrange equation for minimizing
Z
J(u) =
|∇u(x)|q dx
Ω
subject to the constraint
Z
|u(x)|r dx = 1
H(u) =
Ω
where q, r > 1.
62. An interpolation inequality of so-called Landau-Kolmogorov type states that
||u0 ||2L2 (a,b) ≤ C||u||L2 (a,b) ||u00 ||L2 (a,b)
for some constant C independent of u ∈ H 2 (a, b).
a) Use this to show that
21
Z b
00
2
2
(u (x) + u(x) ) dx
a
H 2 (a, b).
is an equivalent norm on
b) If f ∈ L2 (0, 1) we say that u is a weak solution of the fourth order problem
u0000 + u = f
0<x<1
u00 (0) = u000 (0) = u00 (1) = u000 (1) = 0
if u ∈ H 2 (0, 1) and
Z
1
00
Z
00
0
1
f (x)ζ(x) dx
(u (x)ζ (x) + u(x)ζ(x)) dx =
for all ζ ∈ H 2 (0, 1)
0
Discuss why this is a reasonable definition.
c) Use the Lax-Milgram Theorem to prove that the problem in part b) has a weak solution.
63. Let f and g be in L2 (0, 1). Use the Lax-Milgram Theorem to prove there is a unique weak
solution {u, v} ∈ H01 (0, 1) × H01 (0, 1) to
−u00 + u + v 0 = f
−v 00 + v + u0 = g,
where u(0) = v(0) = 0, u(1) = v(1) = 0. (Hint: Start by defining the bilinear form
Z
a((u, v), (φ, ψ)) =
0
on H01 (0, 1) × H01 (0, 1).)
1
(u0 φ0 + uφ + v 0 φ + v 0 ψ 0 + vψ + u0 ψ) dx