Sign-changing solutions to Schiffer's overdetermined problem on wavy cylinder

Abstract

In this paper, we prove the existence of $k$ families of smooth unbounded domains ${\Omega }_s\subset {\mathbb{R}}^{N+1}$ with $N\ge 1$, where \begin{equation} Ω_s=\left\{(x,t)\in \mathbb{R}^N\times \mathbb{R}:\vert x\vert<1+s\cos \left(\frac{2π}{T(s)}t\right)+s w_s\left(\frac{2π}{T(s)}t\right)\right\},\nonumber \end{equation} such that \begin{equation} -Δu=λu\,\, \text{in}\,\,Ω, \,\, \partial_νu=0,\,\,u=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} admits a bounded sign-changing solution with exactly $k+1$ nodal domains. These results can be regarded as counterexamples to the Schiffer conjecture on unbounded domain. These results also indicate that there exist non-spherical unbounded regions without Pompeiu property. Our construction shows that the condition "$\partial \Omega$ is homeomorphic to the unit sphere" is necessary for Williams conjecture to hold. In addition, these conclusions may have potential applications in remote sensing or CT.

We find that the proof of our main Theorem 1.1 and Theorem 1.2 are wrong, where the essential reason is that the linearized operator in 23 page is not order 1 elliptic. Thus we wish to withdraw all versions of this paper