On Dirichlet eigenvalues of measure differential equations with indefinite weight measures

In this paper, we study Dirichlet eigenvalue problem of the second order measure differential equation with an indefinite weight measure. The main result is a complete description on eigenvalues of the problem. To obtain such a result, we will first establish a sufficient and necessary condition on the existence of the first positive and negative eigenvalues of the problem. Secondly, we will give a fully description on eigenvalues of the problem when the indefinite weight measures are singular measures. Thirdly, by constructing approximating measures and developing some convergence result on eigenvalues, we will prove the main result. Finally, we will propose some optimization problems on the first positive eigenvalue.