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