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 \(n\ge 3\) $n\ge 3$ with a non-empty smooth boundary \(\partial M\) $\partial M$ . We show that for any finite increasing sequence of real numbers \(0<a_1<a_2 \le a_3 \le \cdots \le a_N\) $0<a_1<a_2\le a_3\le \cdots \le a_N$ and any positive number V, there exists a Riemannian metric g on M such that \(\textrm{Vol}(M,g)=V\) $\text{Vol}(M,g)=V$ and \(\lambda ^\mathcal {D}_k(M,g)=a_k\) ${\lambda }_k^{\mathcal{D}}(M,g)=a_k$ for any integer \(1 \le k \le N\) $1\le k\le N$ .