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

TL;DR

This paper investigates the asymptotic behavior of principal eigenvalues and eigenfunctions of certain elliptic operators with large advection, providing new insights under minimal regularity assumptions and numerical validation.

Contribution

It offers a combined numerical and analytical study of eigenfunction limits and derives new asymptotic results for eigenvalues in one-dimensional cases with minimal regularity.

Findings

Eigenfunctions approximate 1 and their derivatives approximate 0 as s→∞ under certain conditions.

Derived new asymptotic behavior of principal eigenvalues in one-dimensional cases with minimal regularity.

Oscillatory properties of m(x) can significantly influence the eigenvalues and eigenfunctions' asymptotics.

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, $\varphi_s$ approximates $1$ and $\varphi_s'$ 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 $(\lambda_s,\varphi_s)$ as $s\uparrow +\infty$.