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

Let $\mathcal{L}=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s$ with $s\in (0,1)$ on a bounded $C^{1,1}$ domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, under a partition of the exterior ${\mathbb{R}}^n\setminus \overline{\mathrm{\Omega }}$ 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 ${\lambda }_1^{\mathrm{D}\mathrm{i}\mathrm{r}}-{\lambda }_1(D)$ by integrals of the Dirichlet kernel over $N$ plus a boundary term and a standard fractional tail. When $D$ dissipates and $0<s<\frac{1}{2}$, we bound ${\lambda }_1(D)$ by integrals of the geometric kernel over $D$ and the same tail; for $s\ge \frac{1}{2}$ we give a separated-Dirichlet variant. The proofs use only the weak formulation, the basic spectral theory for the mixed problem, $L^{\mathrm{\infty }}$ 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 $(n,s,\mathrm{\Omega })$ and, in the separated-Dirichlet variant, also on a fixed separation $\delta >0$.