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

Let $ \mathcal{L} = -\Delta+(-\Delta)^s $ with $ s\in(0, 1) $ on a bounded $ C^{1, 1} $ domain $ \Omega\subset \mathbb{R}^n $, under a partition of the exterior $ \mathbb{R}^n\!\setminus\!\overline\Omega $ 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 $ \lambda_1(D) $ along families of partitions in which the Neumann set $ N $ or the Dirichlet set $ D $ dissipates. When $ N $ dissipates, we bound $ \lambda_1^{\mathrm{Dir}}-\lambda_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 < \tfrac12 $, we bound $ \lambda_1(D) $ by integrals of the geometric kernel over $ D $ and the same tail; for $ s\ge\tfrac12 $ we give a separated-Dirichlet variant. The proofs use only the weak formulation, the basic spectral theory for the mixed problem, $ L^\infty $ bounds for principal eigenfunctions, and two cross-testing identities, with all constants and dependencies made explicit. Consequences include quantitative continuity of $ \lambda_1 $ under weak set convergence and a controlled shift of asymptotically linear bifurcation thresholds. All constants depend only on $ (n, s, \Omega) $ and, in the separated-Dirichlet variant, also on a fixed separation $ \delta > 0 $.