A contractible Schiffer counterexample on the half-sphere

TL;DR

The paper constructs contractible domains on the sphere, contained in the upper hemisphere, that admit a nontrivial Neumann eigenfunction which is constant on the boundary, providing a contractible Schiffer-type counterexample on $\mathbb{S}^2$. The authors locate a transversal intersection between a zonal Neumann eigenvalue and a nonzonal Dirichlet eigenvalue on spherical caps via computer-assisted interval arithmetic for Legendre functions, and then execute a Crandall–Rabinowitz bifurcation from geodesic disks to produce a local family of perturbed domains and eigenfunctions. They develop an anisotropic Hölder space framework to rigorously carry out the bifurcation analysis and establish analyticity of the bifurating objects, with the computer-assisted steps oriented around a rigorous enclosure of spectral data and a Poincaré–Miranda certificate of transversal intersection. The results yield a concrete, sign-changing solution to a relaxed overdetermined problem on $\mathbb{S}^2$ and illuminate the curvature-driven mechanism enabling Schiffer-type flexibility on the sphere, with implications for Pompeiu-type questions.

Abstract

We show the existence of a family of nontrivial smooth contractible domains on the sphere that admit Neumann eigenfunctions of the Laplacian which are constant on the boundary. These domains are contained on the half-sphere, in stark contrast with the rigidity literature for Serrin-type problems. The proof relies on a local bifurcation argument around the family of geodesic disks centered at the north pole. We combine the use of anisotropic Hölder spaces for the functional setting with computer-assisted techniques to check the bifurcation conditions.

Figures (3)

• Figure 1: Schematic picture of the spherical cap $\Omega_{a_\star}$ (blue) and our perturbed domain $\Omega^{b_s}$ (green). Our spherical coordinates are $(\theta, \phi)$ and our caps are centered around the north pole $\rm N$.
• Figure 2: The plot shows the curves $(\lambda, a)$ for which $\lambda$ is an eigenvalue of $\Omega_a$. Blue corresponds to the cases where $\lambda$ is the eigenvalue of a zonal Neumann eigenfunction. Red corresponds to the cases where $\lambda$ is the eigenvalue of a Dirichlet eigenfunction with mode $\ell = 8$. We observe that at $a = -1$ we recover the sphere spectrum $n(n+1)$. Our intersection of interest happens at $a = 0.477\ldots$ and $\lambda = 154.19\ldots$, and we can visually see it is transversal on the right plot. The values of $\lambda^l$, $\lambda^u$, $a^l$, $a^u$ are given in \ref{['eq:rho_lambda_box']}--\ref{['eq:a_nu_box']}.
• Figure 3: Crossings between zonal Neumann eigenvalues (red) and non-zonal Dirichlet ones (blue) with angular mode $\cos (\ell_{\text{Dir}} \phi )$, for different domains. The horizontal axis denotes the eigenvalue $\lambda$. The vertical one is the height a of geodesic ball centered around the north pole for all the unit sphere cases. In the case of Figure \ref{['fig:b']}, we use polar coordinates $(r, \varphi)$ on the hyperbolic plane so that $\Delta_{\mathbb H^2} = \frac{\partial^2}{\partial r^2} + \coth(r) \frac{\partial}{\partial r} - \frac{1}{\sinh (r)^2} \frac{\partial^2}{\partial \phi^2}$ and the vertical axis of Figure \ref{['fig:b']} denotes the radius $r$ of our geodesic ball centered at $0$.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof