Math 401: Assignment 9 (Due Mon., Mar. 26 at the start of class)
1. (Euler-Lagrange equations) Let D ⊂ Rn be a bounded domain. For the following
functionals and boundary conditions, find the Euler-Lagrange equation and boundary
conditions satisfied by extremizers (maximizers or minimizers):
(a)
Z
I(u) =
2
2
p(x)|∇u| + q(x)u − r(x)u dx +
D
Z
f (x)u dS(x)
∂D
where p,q, and r are given functions on D and f is a given function on ∂D.
(b)
Z I(u) =
D
1
1 2
2
2
|∇u| + (u − 1) dx.
2
4
(c) The class of functions is
H = {u : D×[0, T ] → R | u(x, t) = g(x) for x ∈ ∂D, u(x, 0) = u0 (x), u(x, T ) = u1 (x)}
where g is a given function on ∂D, and u0 and u1 are given functions on D. The
functional is
!
2
Z Z
1 T
∂u
2
I(u) =
dx dt.
− |∇x u|
2 0 D
∂t
Can you name this Euler-Lagrange equation?
2. (Dirichlet vs. Neumann (natrual) BCs) For functions f (x) on [−1, 1], define the
functional
Z 1
1 0
I(f ) :=
(f (x))2 + xf (x) dx.
−1 2
(a) Find and solve the Euler-Lagrange equation for the problem of minimizing I(f )
among smooth functions with f (−1) = f (1) = 0.
(b) Find and solve the Euler-Lagrange equation for the problem of minimizing I(f )
among all smooth functions.
(c) Compare the minimum values of I for these two problems.
(d) (a bit more challenging) Show that in each case your solution(s) are really
minimizers.
3. (An isoperimetric problem) Consider the class of ”star-shaped” regions D in the
plane, which can be described in polar coordinates as
D = { (r, θ) | 0 ≤ θ < 2π, 0 ≤ r ≤ f (θ) }
for some function f (θ) > 0. We are looking for the region which has maximum area
among those star-shaped regions with a fixed perimeter P (an isoperimetric problem).
(a) Write down a constrained variational problem describing this situation.
(b) Find the Euler-Lagrange equation for this constrained problem.
1
(c) Show that a disk solves the Euler-Lagrange equation.
(d) Consider instead the problem of minimizing the area with fixed perimeter. Does
this problem have a solution? Is the Euler-Lagrange equation any different?
Explain.
4. (The shape of a hanging cable. Consider an (idealized) cable of length L > 2,
and of constant density, hanging between the fixed points (−1, 0) and (1, 0), so that
its location in the plane is described by the graph (x, −u(x)) of the function u(x) ≥ 0,
−1 ≤ x ≤ 1, u(±1) = 0. The cable will hang so as to minimize its (gravitational
potential) energy
Z 1p
E(u) =
1 + (u0 (x))2 u(x)dx
−1
while preserving its length
Z
1
L(u) =
p
1 + (u0 (x))2 dx = L.
−1
(a) Write down the constrained variational problem describing this situation.
(b) Find the Euler-Lagrange equation for this constrained problem, simplifying as
much as possible.
(c) Show that a catenary, u(x) = A − B cosh(x/B), for a suitable choice of A and B,
solves the Euler-Lagrange equation, and satisfies the length constraint and the
boundary conditions.
Mar. 19
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