Differential equations defined by Kreĭn-Feller operators on Riemannian manifolds

Abstract

We study linear and semi-linear wave, heat, and Schr\"odinger equations defined by Kre\u{\i}n-Feller operator $-\Delta_\mu$ on a complete Riemannian $n$-manifolds $M$, where $\mu$ is a finite positive Borel measure on a bounded open subset $\Omega$ of $M$ with support contained in $\overline{\Omega}$. Under the assumption that $\underline{\operatorname{dim}}_{\infty}(\mu)>n-2$, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition $\dim_(\mu)>n-2$ and provide examples of measures on $\mathbb{S}^2$ and $\mathbb{T}^2$ that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on $\mathbb{S}^1$