So we are given the following:
- Operator $A$ with $Au=-u''$;
- $u \in D_A = \{u\colon[a,b]\rightarrow R,u\in C^2([a,b]),u(a)=u(b)=0\}$;
- $D_A$ is dense in $L^2((a,b))$.
Find the minimum value that is possible for the maximum eigenvalue of $A$.
So I have proven that:
- operator $A$ is self-adjoint
- Operator $A$ is positively defined that is $\langle Ax ,x\rangle >0$. But afterwards I'm really not sure how to handle the problem.