P. 8 of Chavel's book Eigenvalues in Riemannian Geometry lists several problems, one being:
Later, the author states the following two results:
Question 1: Is the assumption that $\partial M\neq\emptyset$ essential?
Question 2: Do the above two results still hold true if $U$ is, for example, an open submanifold of a closed Riemannian manifold $N$?
Comments: For Q2, it seems to me that since $L^2(U)\cong L^2(\overline U)$, there still exists an orthonormal basis for $L^2(U)$ consisting of eigenfunctions of $\Delta|_U$, and that the associated eigenvalues satisfy Weyl's law. On the other hand, there could be additional elements in the spectrum of $\Delta|_U$ in the absence of a boundary condition.