A Counterexample for the Principal Eigenvalue of An Elliptic Operator with Large Advection

2023/11/11 by Xueli Bai, Xin Xu, Bai, Xueli +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2311.06475

openalex publication_date 2023/11/11 · openalex created_date 2023/11/15 · openalex updated_date 2026/08/01

Abstract

There are numerous studies focusing on the convergence of the principal eigenvalue λ(s) as s→+∞ corresponding to the elliptic eigenvalue problem -Δφ(x)-2sv⋅∇φ(x)+c(x)φ(x)=λ(s)φ(x), x∈ Ω, where Ω is a bounded domain and the advection term v under some certain restrictions. In this paper, we construct an infinitely oscillating gradient advection term v=∇ m(x)∈ C¹(Ω) such that the principal eigenvalue λ(s) does not converge as s→+∞. As far as we know, this is the first result that guarantee the non-convergence of the principal eigenvalue.

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