Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder

[2307.11441] Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2307.11441 (math) [Submitted on 21 Jul 2023 ( v1 ), last revised 16 Aug 2026 (this version, v8)] Title: Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder Authors: Guowei Dai , Yingxin Sun , Yong Zhang View a PDF of the paper titled Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder, by Guowei Dai and 2 other authors View PDF HTML (experimental) Abstract: Fix $R\in(0,1)$ and $N\geq2$, and let $\lambda_k$ be the $k$th eigenvalue of the radial Dirichlet problem on the annulus $A_R:=\{x\in\mathbb R^N:R<|x|<1\}$. For every $k\geq1$ we construct a smooth local family of rotationally symmetric, axially periodic perturbations of the hollow cylinder $A_R\times\mathbb R$ for which \begin{equation} -\Delta u=\lambda u\quad\hbox{in }\Omega,\qquad u=0\quad\hbox{on }\partial\Omega_{\mathrm{out}}\cup\partial\Omega_R,\qquad \partial_\nu u=\hbox{constant}\quad\hbox{on }\partial\Omega_{\mathrm{out}} \nonumber \end{equation} admits a nonradial periodic solution. The branch bifurcates from the $k$th radial eigenfunction. If $k\geq2$, the solution is sign-changing: it has precisely $k-1$ smooth nested nodal hypersurfaces and hence exactly $k$ nodal domains. We also give an explicit connected-strip analogue when $N=1$. The proof is based on a mean-free Dirichlet-to-Neumann operator. A Sturm--Liouville Weyl function yields a unique simple crossing in the first spectral gap, so that the Crandall--Rabinowitz theorem produces a smooth local branch. This gives a partially overdetermined construction on an unbounded hollow domain with disconnected complement; it lies outside the positivity and connected-complement hypotheses of the Berestycki--Caffarelli--Nirenberg conjecture. Comments: 23pp Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2307.11441 [math.AP] (or arXiv:2307.11441v8 [math.AP] for this version) https://doi.org/10.48550/arXiv.2307.11441 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yong Zhang [ view email ] [v1] Fri, 21 Jul 2023 09:05:05 UTC (305 KB) [v2] Fri, 11 Aug 2023 14:08:02 UTC (308 KB) [v3] Sun, 20 Aug 2023 13:15:42 UTC (308 KB) [v4] Fri, 8 Sep 2023 07:15:15 UTC (307 KB) [v5] Tue, 12 Nov 2024 12:10:05 UTC (312 KB) [v6] Mon, 23 Mar 2026 11:33:30 UTC (307 KB) [v7] Fri, 17 Jul 2026 04:19:03 UTC (56 KB) [v8] Sun, 16 Aug 2026 03:35:02 UTC (35 KB) Full-text links: Access Paper: View a PDF of the paper titled Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder, by Guowei Dai and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2023-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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