Bifurcating domains for an overdetermined eigenvalue problem in cylinders

2025/12/18 by Lian, Yuanyuan, Pacella, Filomena, Sicbaldi, Pieralberto
Computer Science · Mathematics · #35B32 #35G15 #35N25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2512.16319

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

Abstract

We study an overdetermined eigenvalue problem for domains Ω contained in the half-cylinder Σ=ω× (0, +∞), based on a bounded regular domain ω⊂ ℝ^N-1. It is easy to see that in any bounded cylinder Ω_t=ω× (0, t), t > 0, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains Ω⊂ Σ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains Ω_tj at the values t_j = \fracπ2√(σ_j) where σ_j (j≥ 1) is a simple Neumann eigenvalue of the Laplace operator on ω⊂ ℝ^N-1. The solutions can be reflected with respect to ω to generate nontrivial solutions in a cylinder.

Citations

• Stability and asymptotic behaviour of one-dimensional solutions in cylinders
• Nontrivial solutions to the relative overdetermined torsion problem in a cylinder
• Energy stability for a class of semilinear elliptic problems
• Sign-changing solution for an overdetermined elliptic problem on unbounded domain
• Overdetermined problems and relative Cheeger sets in unbounded domains
• Overdetermined elliptic problems in onduloid-type domains with general nonlinearities
• Isoperimetric cones and minimal solutions of partial overdetermined\n problems
• Overdetermined problems and constant mean curvature surfaces in cones
• Solutions to overdetermined elliptic problems in nontrivial exterior\n domains
• New examples of extremal domains for the first eigenvalue of the\n Laplace-Beltrami operator in a Riemannian manifold with boundary
• Bifurcating extremal domains for the first eigenvalue of the Laplacian
• A symmetry problem in potential theory

Related

• Bifurcating extremal domains for the first eigenvalue of the Laplacian
• Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains
• Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction
• Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder
• Overdetermined elliptic problems in onduloid-type domains with general nonlinearities