EXERCISE 1 Consider the eigenvalue problem for the bi-L

EXERCISE 1
Consider the eigenvalue problem for the bi-Laplacian on the interval [-1 1]:
We look for eigenvalues i.e. real for which the problem (1) has a nontrivial solution. These eigenvalues can be enumerated as
Find numerically the first few (say and ) and write down an approximate(asymptotic) formula for as .
EXERCISE 2
Let (square of side length 2). Consider the eigenvalue problem for the bi-Laplacian on
The problem (2) has eigenvalues which can be enumerated as
The aim of this part of the project is to estimate
There is no analytic formula for . However it is known that
Where
and
A convenient way of finding the infimum (3) is to make use of the eigenfunctions of the 1-dimensional problem (1). Let be the span of functions with . Note that is a vector space of dimension . It is known that formula (3) can be rewritten n the form
Your task is to use formula (4) to provide a numerical estimate for
Attachments: