Symmetry and Qualitative \& Quantitative Stability for a Class of Overdetermined Problems in C-GNP Domains with Source Supported in the Core

We introduce a unified geometric framework for domains satisfying a geometric normal property (C-GNP) relative to a strictly convex set $C$. Under the fundamental assumption that the source $f$ is supported within the core $C$, we establish the stability of superlevel sets for elliptic equations and prove a rigid symmetry result for a classical Serrin-type problem via a method that avoids moving planes, relying instead on geometric monotonicity and the Hopf boundary lemma. We then extend this analysis to a coupled biharmonic overdetermined problem $\mathrm{P}(\kappa)$ with source supported in the core. Using the compactness properties of the C-GNP class and the stability of thickness functions under Hausdorff convergence, we prove a qualitative stability theorem: if the overdetermined condition is approximately satisfied in $L^2$ norm, the domain converges in the Hausdorff sense to the unique ball solution. Furthermore, we establish a quantitative stability estimate: there exists a constant $C$ such that $$\rho_e - \rho_i \le C \big\| |\nabla u| |\nabla v| - \kappa \big\|_{L^2(\partial \Omega)}^{\tau_N},$$ with $\tau_2 = 1$, $\tau_3$ arbitrarily close to 1, and $\tau_N = 2/(N-1)$ for $N \ge 4$ in the general case. For convex domains, we improve the exponent to $\tau_N = 4/(N+1)$ via a weighted Reilly identity. The proof relies on Reilly-type integral identities adapted to the coupled system and Hardy-Poincar\'e inequalities tailored to the geometry.