Differential equations defined by Kre\u{\i}n-Feller operators on Riemannian manifolds

TL;DR

This paper investigates wave, heat, and Schrödinger equations on Riemannian manifolds defined by Kreeller operators associated with measures, establishing existence and uniqueness of solutions under certain dimensional conditions.

Contribution

It introduces a framework for analyzing PDEs on manifolds with measures satisfying specific dimensional criteria, extending classical results to more general measure-supported operators.

Findings

Existence and uniqueness of weak solutions for equations with Kreeller operators.

Identification of the critical measure dimension condition im_()>n-2.

Examples of measures on ^2 and ^2 with the required properties.

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 $\mathbb{S}^1$