Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be determined, say depending on a small parameter $\epsilon$, then it is possible to order eigenvalues and eigenfunctions so that they are smooth functions of $\epsilon$ and spatial variables. At first this seemed a very natural result, but the total scarcity of references suggests otherwise, and I am starting to think that this kind of result is rather difficult, if one wants to achieve some generality. Has anyone encountered any result in this direction?
This is true, as long as your domain depends smoothly upon one real parameter. Say that you are insterested in the $n$ first eigenvalues. Using a Lyapunov-Schmidt procedure, you may reduce to the situation of an $n\times n$ symmetric matrix $S(\epsilon)$. Then look at Kato's book in the Grundlehren series.
If instead your domain depends on two or more parameters, the matrix $S$ will depend on several variables, and the eigenvalues will not be smooth functions, unless they remain simple.