Math 401: Assignment 8 (Due Mon., Mar. 19 at the start of class)
1. Consider the following Neumann ODE eigenvalue problem on D = [0, 1]:
d2
(− dx
0<x<1
2 + x)φ = λφ
,
φ0 (0) = 0 = φ0 (1)
a small, positive number.
(a) Find the maximum and minimum values of x on D, and use this knowledge to
find upper and lower bounds for the Neumann eigenvalues.
(b) Find a (somewhat better) upper bound for the first eigenvalue using the variational principle. Hint: take, as a trial function, the first Neumann eigenfunction
of the “unperturbed problem” (the above problem but with = 0).
(c) (more challenging!) Find a (somewhat better) upper bound for the second
eigenvalue using the max-min principle. Hint: you will need a 2-parameter family
of trial functions (to satisfy an orthogonality condition) – try a linear combination
of the first two Neumann eigenfunctions of the unperturbed problem.
2. Obtain upper and lower bounds for the first eigenvalue λ1 of the problem
−((x2 + 1)y 0 )0 = λy,
1 ≤ x ≤ 2,
y(1) = y(2) = 0
(a) by comparing it with a constant-coefficient problems.
(b) by comparing it with the (explicitly solvable) problem
−(x2 y 0 )0 = µy,
1 ≤ x ≤ 2,
y(1) = y(2) = 0.
(c) Which bounds are better?
3. (Eigenvalues on disks and ellipses) Consider the Dirichlet eigenvalue problem
−∆φ = λφ
x∈D
.
φ=0
x ∈ ∂D
(a) Let D = Da be the disk of radius a in the plane. Find (and compare) upper
2
2
2
bounds for the first
p eigenvalue λ1 by using the test functions u(x) = a − x1 − x2
2
2
and v(x) = a − x1 + x2 in the variational principle.
(b) Again, for D = Da , determine all the eigenvalues and eigenfunctions. (Hint:
separate variables in polar coordinates. The solutions of the ODE vrr + vr /r +
(λ − n2 /r2 )v = 0 which
√ are finite at r = 0 are multiples of the Bessel function of
order n: v(r) = AJn ( λr). The Bessel function Jn has infinitely many positive
zeros (points α where Jn (α) = 0): denote them by αn,k , with k = 1, 2, 3, . . ., with
αn,1 < αn,2 < αn,3 < · · · . Your answer will be expressed in terms of these zeros.)
Which is the first eigenvalue? (Hint: for n < m, αn,k < αm,k .) Compare your
upper bound from part (a) with the true first eigenvalue, using α0,1 ≈ 2.40483.
(c) Now let D = Da,b be the region enclosed by the ellipse x2 /a2 + y 2 /b2 = 1 (a < b).
Find upper and lower bounds for the first Dirichlet eigenvalue for Da,b by using
the result of part (b), for appropriately sized disks.
Mar. 15
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