Eigenvalues of the Laplacian on a square
What are the solutions to the equation
uxx + uyy = u
on the unit square with the requirement that u(x, y) = 0 on the boundary?
It's easy to see that the functions
u(x, y) = sin(mx) sin(ny)
are solutions with
= (m^2 + n^2)^2
for non-negative integersm andn. It's not so obvious that these are the only solutions, but we'll take that on faith.
The previous post looked at Polya's bounds on the eigenvalues of the Laplace operator for a general region D, with only the requirement that copies of D can tile the plane without overlapping. Surely squares satisfy this requirement, so the problem in this post is a special case of the problem in the previous post. So how do they compare?
Polya gives lower bounds on thekth eigenvalue, so how do we order the numbers of the form (m^2 + n^2)?
There's a theorem for counting the number of numbersn up tox wheren is the sum of two squares, the Landau-Ramanujan theorem. But it seems to contradict Polya's bounds. That's because Landau-Ramanujan only counts eachn once, but eigenvalues need to be counted multiple times if a number can be written as a sum of squares multiple ways.
For example, 25^2 should count as an eigenvalue four times, corresponding to (m, n) = (5, 0), (0, 5), (3, 4), and (4, 3).
About how large is thekth eigenvalue? If non-negative integers m and n satisfy
(m^2 + n^2)^2 x
then (m,n) is inside a circle of radius x/. For largex, the number of such pairs is approximately the area of a circle of radius x/ in the first quadrant, which isx / 4. So thekth eigenvalue is approximately 4k, which matches Polya's lower bound of 4k for a region of area 1.
The post Eigenvalues of the Laplacian on a square first appeared on John D. Cook.