Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum.

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold M of dimension n ≥ 3 with a non-empty smooth boundary ∂ M . We show that for any finite increasing sequence of real numbers 0 < a 1 < a 2 ≤ a 3 ≤ ⋯ ≤ a N and any positive number V, there exists a Riemannian metric g on M such that Vol (M , g) = V and λ k D (M , g) = a k for any integer 1 ≤ k ≤ N . [ABSTRACT FROM AUTHOR]