Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection
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| Format: | Preprint |
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2025
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| _version_ | 1866909977083379712 |
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| author | Guo, Yujin Lou, Yuan Zhang, Hongfei |
| author_facet | Guo, Yujin Lou, Yuan Zhang, Hongfei |
| contents | In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -εΔϕ-2α\nabla m(x)\cdot\nabla ϕ+V(x)ϕ&=λϕ \ \text{in}\ \ Ω,\\ ϕ&=0\ \ \hbox{on}\ \ \partialΩ, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $Ω\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partialΩ\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $ε>0$ and $α>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\barΩ)$, $V (x)\in C^γ(\barΩ)~(0<γ<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(λ, ϕ)$ for (0.1) as $α\rightarrow\infty$, which display the visible effect of the large advection on $(λ, ϕ)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection Guo, Yujin Lou, Yuan Zhang, Hongfei Analysis of PDEs In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -εΔϕ-2α\nabla m(x)\cdot\nabla ϕ+V(x)ϕ&=λϕ \ \text{in}\ \ Ω,\\ ϕ&=0\ \ \hbox{on}\ \ \partialΩ, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $Ω\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partialΩ\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $ε>0$ and $α>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\barΩ)$, $V (x)\in C^γ(\barΩ)~(0<γ<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(λ, ϕ)$ for (0.1) as $α\rightarrow\infty$, which display the visible effect of the large advection on $(λ, ϕ)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems. |
| title | Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.22918 |