Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection
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arXiv
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| Natura: | Preprint |
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2025
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| author | Cano-Casanova, S. López-Gómez, J. Molina-Meyer, M. |
| author_facet | Cano-Casanova, S. López-Gómez, J. Molina-Meyer, M. |
| contents | In this paper we study, both numerically and analytically, the asymptotic behavior of the principal eigenfunction of \eqref{1.1}, normalized by \eqref{1.2}, as $s\uparrow +\infty$. Based on the numerical computations of this paper, we can prove that, under condition (Hm) bellow, $φ_s$ approximates $1$ and $φ_s'$ approximates $0$, uniformly in $[-1,1]$, as $s\uparrow +\infty$. As a byproduct of this result, we can derive the asymptotic behavior of the principal eigenvalue in a one-dimensional situation not previously covered by \cite{ChLo} and \cite{PeZh}, as we are working under minimal regularity assumptions on $m(x)$. A recent result of \cite{BWZ} shows that the principal eigenvalue might oscillate as $s\uparrow +\infty$ if $m(x)$ is highly oscillatory. Thus, the oscillatory and regularity properties of $m(x)$ might severely affect the asymptotic behavior of $(λ_s,φ_s)$ as $s\uparrow +\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection Cano-Casanova, S. López-Gómez, J. Molina-Meyer, M. Analysis of PDEs Classical Analysis and ODEs 34B09, 34D15, 34L15 In this paper we study, both numerically and analytically, the asymptotic behavior of the principal eigenfunction of \eqref{1.1}, normalized by \eqref{1.2}, as $s\uparrow +\infty$. Based on the numerical computations of this paper, we can prove that, under condition (Hm) bellow, $φ_s$ approximates $1$ and $φ_s'$ approximates $0$, uniformly in $[-1,1]$, as $s\uparrow +\infty$. As a byproduct of this result, we can derive the asymptotic behavior of the principal eigenvalue in a one-dimensional situation not previously covered by \cite{ChLo} and \cite{PeZh}, as we are working under minimal regularity assumptions on $m(x)$. A recent result of \cite{BWZ} shows that the principal eigenvalue might oscillate as $s\uparrow +\infty$ if $m(x)$ is highly oscillatory. Thus, the oscillatory and regularity properties of $m(x)$ might severely affect the asymptotic behavior of $(λ_s,φ_s)$ as $s\uparrow +\infty$. |
| title | Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection |
| topic | Analysis of PDEs Classical Analysis and ODEs 34B09, 34D15, 34L15 |
| url | https://arxiv.org/abs/2508.16108 |