@article{Panraksa2025, 
author = {Chatchawan Panraksa},
title = {Quantitative stability of the principal eigenvalue for mixed local–nonlocal operators under dissipating boundary partitions},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {28115-28128},
keywords = {mixed local–nonlocal operator, fractional Laplacian, mixed boundary conditions, principal eigenvalue, stability},
url = {https://www.sciopen.com/article/10.3934/math.20251236},
doi = {10.3934/math.20251236},
abstract = {Let        L    =  −  Δ  +  (  −  Δ      )    s   with    s  ∈  (  0  ,  1  ) on a bounded        C          1      ,      1       domain    Ω  ⊂            R        n  , under a partition of the exterior              R        n      ∖        Ω    ¯   into disjoint open sets    D (Dirichlet) and    N (nonlocal Neumann). Building on the mixed local–nonlocal framework, we obtain explicit, provable upper bounds for the variation of the principal eigenvalue        λ    1    (  D  ) along families of partitions in which the Neumann set    N or the Dirichlet set    D dissipates. When    N dissipates, we bound        λ    1                  D        i        r              −      λ    1    (  D  ) by integrals of the Dirichlet kernel over    N plus a boundary term and a standard fractional tail. When    D dissipates and    0  <  s  <            1      2      , we bound        λ    1    (  D  ) by integrals of the geometric kernel over    D and the same tail; for    s  ≥            1      2       we give a separated-Dirichlet variant. The proofs use only the weak formulation, the basic spectral theory for the mixed problem,        L    ∞   bounds for principal eigenfunctions, and two cross-testing identities, with all constants and dependencies made explicit. Consequences include quantitative continuity of        λ    1   under weak set convergence and a controlled shift of asymptotically linear bifurcation thresholds. All constants depend only on    (  n  ,  s  ,  Ω  ) and, in the separated-Dirichlet variant, also on a fixed separation    δ  >  0.}
}