Panraksa C.Mahidol University2025-12-062025-12-062025-01-01Aims Mathematics Vol.10 No.12 (2025) , 28115-28128https://repository.li.mahidol.ac.th/handle/123456789/113408Let L = −∆ + (−∆)^s with s ∈ (0, 1) on a bounded C^1,1 domain Ω ⊂ Rⁿ, under a partition of the exterior Rⁿ\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^Dir<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>¹, we bound λ<inf>1</inf>(D) by integrals of the geometric kernel over D and the same tail; for s ≥ ¹<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^∞ 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.MathematicsArticleSCOPUS10.3934/math.202512362-s2.0-10502329285824736988