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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.30026 |
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| _version_ | 1866914494089789440 |
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| author | Barkatou, Mohammed |
| author_facet | Barkatou, Mohammed |
| contents | 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 \(\mathrm{P}(κ)\) 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 \[ ρ_e - ρ_i \le C \big\| |\nabla u| |\nabla v| - κ\big\|_{L^2(\partial Ω)}^{τ_N}, \] with \(τ_2 = 1\), \(τ_3\) arbitrarily close to 1, and \(τ_N = 2/(N-1)\) for \(N \ge 4\) in the general case. For convex domains, we improve the exponent to \(τ_N = 4/(N+1)\) via a weighted Reilly identity. The proof relies on Reilly-type integral identities adapted to the coupled system and Hardy-Poincaré inequalities tailored to the geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_30026 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetry and Qualitative \& Quantitative Stability for a Class of Overdetermined Problems in C-GNP Domains with Source Supported in the Core Barkatou, Mohammed Analysis of PDEs 35J25, 35J30, 49Q10, 35B06 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 \(\mathrm{P}(κ)\) 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 \[ ρ_e - ρ_i \le C \big\| |\nabla u| |\nabla v| - κ\big\|_{L^2(\partial Ω)}^{τ_N}, \] with \(τ_2 = 1\), \(τ_3\) arbitrarily close to 1, and \(τ_N = 2/(N-1)\) for \(N \ge 4\) in the general case. For convex domains, we improve the exponent to \(τ_N = 4/(N+1)\) via a weighted Reilly identity. The proof relies on Reilly-type integral identities adapted to the coupled system and Hardy-Poincaré inequalities tailored to the geometry. |
| title | Symmetry and Qualitative \& Quantitative Stability for a Class of Overdetermined Problems in C-GNP Domains with Source Supported in the Core |
| topic | Analysis of PDEs 35J25, 35J30, 49Q10, 35B06 |
| url | https://arxiv.org/abs/2603.30026 |