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Schiffer problem and isoparametric hypersurfaces. (English) Zbl 0976.35015

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.

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
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