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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.19861 |
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| _version_ | 1866915877537972224 |
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| author | Xu, Xin Zhang, Kexin |
| author_facet | Xu, Xin Zhang, Kexin |
| contents | This paper investigates the asymptotic behavior of the principal eigenvalue $λ(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E}
-Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold $(M,g)$. Assuming $f$ is a Morse function defined on $M$, we find that the limit $\lim\limits_{s\to+\infty} λ(s)$ is determined by the minimum value of the function $c$ over the set of the maximum points of $f$, a result that is independent of the curvature of manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds Xu, Xin Zhang, Kexin Analysis of PDEs This paper investigates the asymptotic behavior of the principal eigenvalue $λ(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold $(M,g)$. Assuming $f$ is a Morse function defined on $M$, we find that the limit $\lim\limits_{s\to+\infty} λ(s)$ is determined by the minimum value of the function $c$ over the set of the maximum points of $f$, a result that is independent of the curvature of manifold. |
| title | Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.19861 |