An overdetermined problem for sign-changing eigenfunctions in unbounded domains

2022/03/29 by Ignace Aristide Minlend, Minlend, Ignace Aristide
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.15492

openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the existence of non-trivial unbounded domains of Ω⊂ ℝ² where the equation - λu_xx -u_tt amp;= u in Ω,u amp;=0 on ∂ Ω,is solvable subject to the conditions (∂ u)/(∂ η) =-1 on ∂ Ω⁺ \textrmand (∂ u)/(∂ η) =+1 on ∂ Ω^-. For every integer m≥ 0, we prove the existence of a family of unbounded domains Ω⊂ ℝ² indexed by 0 \leqslantℓ\leqslant 2m, where the above problem admits periodic sign-changing solutions. The domains we construct are periodic in the first coordinate in ℝ², and they bifurcate from suitable strips.

Related

• Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains
• Sign-changing solution for an overdetermined elliptic problem on unbounded domain
• Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder
• Bifurcating domains for an overdetermined eigenvalue problem in cylinders
• The ∞-eigenvalue problem with a sign-changing weight