Consider the 1D Dirichlet eigenvalue problem for the Laplacian on the interval $\Omega =[0,1]$: \begin{align} u'' + \lambda u & = 0, \quad x \in \Omega, \\ u(0)=u(1) &=0. \end{align}
The eigenvalues of $-d^2/dx^2$ are given by $$ \lambda_n = (n\pi)^2, $$ and the eigenfunctions are given by $$ u_n(x) = \sin(n\pi x). $$
Now I would like to generalize this to two cases:
- $\Omega$ is an arc on a circle: The circle is centered at the origin in $\mathbb{R}^2$ and has radius $R$ with endpoints $a$ and $b$.
- $\Omega$ is a section of the graph of $y(x)=x^2$: Again with some endpoints $a$ and $b$.
The reason for this is that I have very little experience with differential geometry but alot of experience with the Helmholtz equation so I think this would be a good approach to learning how to deal with PDEs on manifolds - it seems like these are some of the simplest generalizations we could consider.
So what do I need to do to obtain the eigenvalues (eigenfunctions) for in cases 1. and 2. above? A list of steps to take would be great so I could then try and work out the details?
Unfortunately I think this will not work as you expect. The curvature of an arc or a section of a graph is only due to the embedding in $\mathbb{R}^2$. The intrinsic curvature is still zero in both cases. So the eigenvalues and eigenfunctions are still exactly the same as for the straight interval (in an arc length parametrization).