Some Results on the Schiffer's Conjecture in R^2

Details Some Results on the Schiffer's Conjecture in R^2 Some Results on the Schiffer's Conjecture in R^2

2011-11-30

arxiv.org/abs/1112.0207v2

Mathematics

Differential Geometry

12 pages, second version, some typos and misprints corrected

Scientific paper

Let \Omega .be an open, bounded domain in the plane with connected and smooth boundary, and \omega .an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue \mu > 0. If the boundary value of \omega .is a nonzero constant along the boundary, denoting 0 = \mu_1(\Omega) < \mu_2(\Omega) <= ... the set of all Neumann eigenvalues for the Laplacian on \Omega, we show that 1) if \mu < \mu_8(\Omega); or 2) if \Omega .is strictly convex and centrally symmetric, \mu < \mu_13(\Omega), then \Omega must be a disk.

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