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

Abstract

Fix $R\in (0,1)$ and $N\ge 2$, 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 $k\ge 1$ 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} -Δu=λu\quad\hbox{in }Ω,\qquad u=0\quad\hbox{on }\partialΩ_{\mathrm{out}}\cup\partialΩ_R,\qquad \partial_νu=\hbox{constant}\quad\hbox{on }\partialΩ_{\mathrm{out}} \nonumber \end{equation} admits a nonradial periodic solution. The branch bifurcates from the $k$th radial eigenfunction. If $k\ge 2$, 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.

23pp