In this paper, we study the minimization of ${\lambda }_1(\mathrm{\Omega })$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets Ω of fixed volume in a Riemannian manifold $(M,g)$. In the Euclidean setting (when $(M,g)=({\mathbb{R}}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

在本文中，我们研究了最小化 ${\lambda }_{\mathrm{1个}}（\mathrm{\Omega }）$，是Laplace-Beltrami运算符的第一个Dirichlet特征值，在黎曼流形中固定体积的开放集Ω的类别内 $（\mathrm{中号}，G）$。在欧几里得设置中（当$（\mathrm{中号}，G）=（{\mathbb{[R}}^{ñ}，Ë）$），著名的Faber-Krahn不等式断言，解决此问题的方法是任何合适体积的球。即使已知或预期具有对称对称的黎曼流形的相似结果，我们也不能期望找到通用流形的显式解$（\mathrm{中号}，G）$。在本文中，我们以与等渗问题类似的方式，研究了在黎曼设置中此频谱形状优化问题的存在性和规则性。我们首先在紧致黎曼流形的背景下给出一个存在性结果，然后通过给出一个反例来讨论非紧致流形的情况。然后，我们将重点放在此问题的正则性理论上，并使用来自自由边界问题理论的工具，我们证明解决方案对维数为5或更高的可能残差集都是平滑的。