Quantitative stability of the principal eigenvalue for mixed local–nonlocal operators under dissipating boundary partitions

Let 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 λ₁(D) along families of partitions in which the Neumann set N or the Dirichlet set D dissipates. When N dissipates, we bound λ^Dir₁ − λ₁(D) by integrals of the Dirichlet kernel over N plus a boundary term and a standard fractional tail. When D dissipates and 0 < s < ₂¹, we bound λ₁(D) by integrals of the geometric kernel over D and the same tail; for s ≥ ¹₂ 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 λ₁ 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.