Eigenfunctions of the Laplacian on singular spaces

Question

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$ on such an $M$ with Dirichlet or Neumann boundary conditions. In such situations, are the eigenfunctions of the Laplacian smooth? It seems to me that this requires some sort of elliptic regularity theory for singular spaces (maybe there is an elliptic regularity theory for manifolds with Lipschitz boundaries). A reference would be highly appreciated.