Shklover, Vladimir E. Schiffer problem and isoparametric hypersurfaces. (English) Zbl 0976.35015 Rev. Mat. Iberoam. 16, No. 3, 529-569 (2000). The author considers the generalization of the Schiffer problem to an arbitrary Riemannian manifold. He proves that if \(\Omega\) has a homogeneous boundary, then the problems (N) and (D) admit solutions (in fact, for infinitely many eigenvalues), but the converse statement is not always true. Further, he shows that in the case of domains with boundaries consisting of isoparametric hypersurfaces (D) the Schiffer conjectures fails. Reviewer: Li Mingzhong (Shanghai) Cited in 1 ReviewCited in 12 Documents MSC: 35J25 Boundary value problems for second-order elliptic equations 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35N05 Overdetermined systems of PDEs with constant coefficients 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces 58J32 Boundary value problems on manifolds Keywords:Schiffer problem; isoparametric hypersurfaces; homogeneous boundary value problem; overdetermined boundary value problem PDFBibTeX XMLCite \textit{V. E. Shklover}, Rev. Mat. Iberoam. 16, No. 3, 529--569 (2000; Zbl 0976.35015) Full Text: DOI EuDML