Prescription of finite Dirichlet eigenvalues and area on surface with boundary

Abstract

In the present paper, we consider Dirichlet Laplacian on compact surface. We show that for a fixed surface with boundary $X$, a finite increasing sequence of real numbers $0<a_1<a_2<\cdots<a_N$ and a positive number $A$, there exists a metric $g$ on $X$ such that for any integer $1\leq k\leq N$, we have $\lambda_k^\mathcal{D}(X,g)=a_k$ and $\mathrm{Area}(X,g)=A$.