Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds

Abstract

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kre\uın-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 ${\underline{\dim }}_{\infty }(\mu )>n-2$, 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 ${\mathbb{S}}^1$

35 pages