The Schiffer problem on the cylinder and on the 2-sphere

  • Mouhamed Moustapha Fall

    African Institute for Mathematical Sciences Senegal, Mbour, Senegal
  • Ignace Aristide Minlend

    University of Douala, Douala, Cameroon
  • Tobias Weth

    Goethe-Universität Frankfurt am Main, Frankfurt, Germany
The Schiffer problem on the cylinder and on the 2-sphere cover
Download PDF

A subscription is required to access this article.

Abstract

We prove the existence of a family of compact subdomains of the flat cylinder for which the Neumann eigenvalue problem for the Laplacian on admits eigenfunctions with constant Dirichlet values on . These domains have the property that their boundaries  have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The functional analytic approach we develop in this paper overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere . By this we disprove a conjecture by Souam [Ann. Global Anal. Geom. 27, 341–354 (2005)].

Cite this article

Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias Weth, The Schiffer problem on the cylinder and on the 2-sphere. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1612