Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence
Abstract
We study the asymptotic behavior as d→0+ of the principal eigenvalue λ(d) of the one-dimensional elliptic problem −dϕ''(x)−2αm'(x)ϕ'(x)+c(x)ϕ(x) =λ(d)ϕ(x), & x∈(a,b), ϕ'(a)=ϕ'(b)=0, where α>0 is fixed, m∈ C1,1([a,b]), and c∈ C([a,b]). We assume that m'=0 only at x0∈(a,b), where m attains a strict minimum. Within this class, we construct a counterexample m∈ C1,1 for which λ(d) has two distinct subsequential limits as d→0+, leading to the non-convergence of λ(d). We then consider another class of C1,1 advection functions for which m'(x)/(x−x0) has positive one-sided limits at x0. For this class, we prove that λ(d) converges as d→0+. Thus, C1,1 regularity alone does not ensure convergence of the principal eigenvalue, even when m has exactly one strict interior critical minimal point.
Related
- Optimal regularity and fine asymptotics for very fast diffusion equations in bounded domains
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BibTeX
@misc{zhang2026regularity,
title = {{Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence}},
author = {Kexin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37259},
eprint = {2610.37259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37259},
}
RIS
TY - UNPB
TI - Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence
AU - Kexin Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37259
UR - https://arxiv.org/abs/2610.37259
AB - We study the asymptotic behavior as d→0^+ of the principal eigenvalue λ(d) of the one-dimensional elliptic problem −dϕ''(x)−2αm'(x)ϕ'(x)+c(x)ϕ(x) =λ(d)ϕ(x), & x∈(a,b), ϕ'(a)=ϕ'(b)=0, where α>0 is fixed, m∈ C^1,1([a,b]), and c∈ C([a,b]). We assume that m'=0 only at x_0∈(a,b), where m attains a strict minimum. Within this class, we construct a counterexample m∈ C^1,1 for which λ(d) has two distinct subsequential limits as d→0^+, 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→0^+. 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.
ER -