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