{
  "abstract": "We study the asymptotic behavior as \\(d\\to0^+\\) of the principal eigenvalue \\(λ(d)\\) of the one-dimensional elliptic problem \\[ \\begin{cases} -dϕ''(x)-2αm'(x)ϕ'(x)+c(x)ϕ(x) =λ(d)ϕ(x), & x\\in(a,b),\\\\ ϕ'(a)=ϕ'(b)=0, \\end{cases} \\] where \\(α>0\\) is fixed, \\(m\\in C^{1,1}([a,b])\\), and \\(c\\in C([a,b])\\). We assume that \\(m'=0\\) only at \\(x_0\\in(a,b)\\), where \\(m\\) attains a strict minimum. Within this class, we construct a counterexample $m\\in C^{1,1}$ for which \\(λ(d)\\) has two distinct subsequential limits as \\(d\\to0^+\\), leading to the non-convergence of $λ(d)$. We then consider another class of $C^{1,1}$ advection functions for which $m'(x)/(x-x_0)$ has positive one-sided limits at $x_0$. For this class, we prove that $λ(d)$ converges as $d\\to0^+$. Thus, \\(C^{1,1}\\) regularity alone does not ensure convergence of the principal eigenvalue, even when \\(m\\) has exactly one strict interior critical minimal point.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37259",
  "author_total": 1,
  "authors": [
    {
      "name": "Kexin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7fd28z1k1hrpa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37259",
  "fulltext_url": null,
  "id": "7fd28z1k1hrpa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37259",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7224094814",
  "pdf_url": "https://arxiv.org/pdf/2610.37259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7fd28z1k1hrpa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence",
  "title_text": "Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/7fd28z1k1hrpa",
  "venue": "arXiv",
  "versions": []
}