Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916350808555520 |
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| author | Ngai, Sze-Man Ouyang, Lei |
| author_facet | Ngai, Sze-Man Ouyang, Lei |
| contents | 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 $\mathbb{S}^1$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds Ngai, Sze-Man Ouyang, Lei Functional Analysis 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 $\mathbb{S}^1$ |
| title | Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2408.04858 |