express it as an eigenvalue problem (sobolev)

Hello

The exercise is given by

Express

$\displaystyle sup_{v \in V \setminus \{0\}} \frac{\int_0^1 v*v}{\int_0^1 v'*v'} $ as an eigenvalue problem where $\displaystyle V \subset W^{1,2}(\]0.1\[) and \forall v \in V : v(0)=0=v(1)$

($\displaystyle W^{k,p}(\]0,1\[)$ denotes the sobolev space on the open intervall (0,1). For those who don't know that just assume V is a subspace of functions on this intervall with boundary values of 0 that have the first order derivative.)

I don't know how to begin this. I have to find something in the way $\displaystyle A*x=b$ where A is a given matrix, b is a given vector and x the solution that has to be found. But how can I do this? Or even better I just have to find a matrix which (absolute) greatest eigenvalue determines this value?

I tried partial integration but it really leads me nowhere... Does anyone have a hint how to solve this?

Regards

Re: express it as an eigenvalue problem (sobolev)

Does noone have a hint or such? How can this be transformed into an eigenvalue problem?