In this paper, we consider an eigenvalue problem defined on a two-patch domain. Our first aim is to show that the two-patch eigenvalue problem is equivalent to the eigenvalue problem of a measure differential equation defined on an one-patch domain. Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously depending on the weight measure in the weak^⁎ topology of the measure space. Our third aim is to solve a minimization problem on principal eigenvalues. Some main results of this paper have interesting relations with population dynamics. We will interpret these results in terms of survival chances and optimal distribution of resources.

在本文中，我们考虑在两色域上定义的特征值问题。我们的第一个目的是证明两补丁特征值问题等同于在一个补丁域上定义的度量微分方程的特征值问题。我们的第二个目标是研究测度微分方程主特征值的存在，并证明在测度空间的弱^⁎拓扑中，主特征值连续地取决于权重测度。我们的第三个目标是解决主特征值的最小化问题。本文的一些主要结果与人口动态具有有趣的关系。我们将根据生存机会和资源的最佳分配来解释这些结果。