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arXiv · 2408.04858

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

Sze-Man Ngai·Lei Ouyang
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Abstract

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kre\uın-Feller operator $-Δ_μ$ on a complete Riemannian $n$-manifolds $M$, where $μ$ is a finite positive Borel measure on a bounded open subset $Ω$ of $M$ with support contained in $\overlineΩ$. Under the assumption that $\underline{\operatorname{dim}}_{\infty}(μ)>n-2$, 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 $\dim_(μ)>n-2$ 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 $\mathbb{S}^1$

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Sze-Man Ngai, Lei Ouyang. 2024-08-09. Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds. https://arxiv.org/abs/2408.04858

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