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

TL;DR

This paper develops a geometric framework to analyze stability and symmetry in overdetermined elliptic problems within C-GNP domains, providing new qualitative and quantitative stability results under source support in the core.

Contribution

It introduces a unified geometric approach to establish stability and symmetry results for overdetermined problems with sources supported in the core, extending previous methods.

Findings

Proved stability of superlevel sets for elliptic equations in C-GNP domains.

Established a qualitative stability theorem showing domain convergence to a ball.

Derived a quantitative stability estimate with explicit exponents depending on dimension.

Abstract

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.