MATH 520 Homework
Spring 2014
56. Let A be an m × n real matrix, b ∈ Rm and define J(x) = ||Ax − b||2 for x ∈ Rn . (Here
||x||2 denotes the 2 norm, the usual Euclidean distance on Rm ).
a) What is the Euler-Lagrange equation for the problem of minimizing J?
b) Under what circumstances does the Euler-Lagrange equation have a unique solution?
c) Under what circumstances will the solution of the Euler-Lagrange equation also be a
solution of Ax = b?
57. Fill in the details of the following alternate proof that there exists a weak solution of the
Neumann problem
−∆u = f
∂u
= 0 x ∈ ∂Ω
∂n
x∈Ω
(NP)
R
(as usual, Ω is a bounded open set in RN ) provided f ∈ L2 (Ω), and Ω f (x) dx = 0:
a) Show that for any > 0 there exists a (suitably defined) unique weak solution u of
−∆u + u = f
x∈Ω
∂u
= 0 x ∈ ∂Ω
∂n
R
b) Show that Ω u (x) dx = 0 for any such .
c) Show that there exists u ∈ H 1 (Ω) such that u → u weakly in H 1 (Ω) as → 0, and u is
a weak solution of (NP).
58. Find the function u(x) which minimizes
1
Z
J(u) =
(u0 (x) − u(x))2 dx
0
among all functions u ∈ H 1 (0, 1) satisfying u(0) = 0, u(1) = 1.
59. The area of a surface obtained by revolving the graph of y = u(x), 0 < x < 1 about the x
axis, is
Z 1
p
J(u) = 2π
u(x) 1 + u0 (x)2 dx
0
Assume that u is required to satisfy u(0) = a, u(1) = b where 0 < a < b.
a) Find the Euler-Lagrange equation for the problem of minimizing this surface area.
b) Show that
p
u(u0 )2
p
− u 1 + (u0 )2
1 + (u0 )2
is a constant function for any such minimal surface (see Proposition 13.37 in text).
c) Solve the first order ODE in part b) to find the minimal surface. Make sure to compute
all constants of integration.