Quantitative stability of the principal eigenvalue for mixed local–nonlocal operators under dissipating boundary partitions
Issued Date
2025-01-01
Resource Type
eISSN
24736988
Scopus ID
2-s2.0-105023292858
Journal Title
Aims Mathematics
Volume
10
Issue
12
Start Page
28115
End Page
28128
Rights Holder(s)
SCOPUS
Bibliographic Citation
Aims Mathematics Vol.10 No.12 (2025) , 28115-28128
Suggested Citation
Panraksa C. Quantitative stability of the principal eigenvalue for mixed local–nonlocal operators under dissipating boundary partitions. Aims Mathematics Vol.10 No.12 (2025) , 28115-28128. 28128. doi:10.3934/math.20251236 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/113408
Title
Quantitative stability of the principal eigenvalue for mixed local–nonlocal operators under dissipating boundary partitions
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Abstract
Let L = −∆ + (−∆)<sup>s</sup> with s ∈ (0, 1) on a bounded C<sup>1,1</sup> domain Ω ⊂ R<sup>n</sup>, under a partition of the exterior R<sup>n</sup>\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<inf>1</inf>(D) along families of partitions in which the Neumann set N or the Dirichlet set D dissipates. When N dissipates, we bound λ<sup>Dir</sup><inf>1</inf> − λ<inf>1</inf>(D) by integrals of the Dirichlet kernel over N plus a boundary term and a standard fractional tail. When D dissipates and 0 < s < <inf>2</inf><sup>1</sup>, we bound λ<inf>1</inf>(D) by integrals of the geometric kernel over D and the same tail; for s ≥ <sup>1</sup><inf>2</inf> we give a separated-Dirichlet variant. The proofs use only the weak formulation, the basic spectral theory for the mixed problem, L<sup>∞</sup> bounds for principal eigenfunctions, and two cross-testing identities, with all constants and dependencies made explicit. Consequences include quantitative continuity of λ<inf>1</inf> 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.