On principal eigenvalues of measure differential equations and a patchy Neumann eigenvalue problem

Abstract

Abstract 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.