Work

Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection

Year: 2026

Type: article

Abstract: 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 $s\uparrow +\infty$.

Source: Mathematics and Computers in Simulation

DOI: 10.1016/j.matcom.2026.03.025

Authors: S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,170

Funders: Ministerio de Ciencia, Innovación y Universidades, Universidad Complutense de Madrid, Ministerio de Ciencia e Innovación

Awards: G1855226563, G300207659