ISSN:
1420-9039
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract In a seminal 1971 paper, James Serrin showed that the only open, smoothly bounded domain in ℝ n on which the positive Dirichlet eigenfunction of the Laplacian has constant (nonzero) normal derivative on the boundary, is then-dimensional ball. The positivity of the eigenfunction is crucial to his proof. To date it is an open conjecture that the same result is true for Dirichlet eigenvalues other than the least. We show that for simply connected, plane domains, the absence of saddle points is a condition sufficient to validate this conjecture. This condition is also sufficient to prove Schiffer's conjecture: the only simply connected planar domain, on the boundary of which a nonconstant Neumann eigenfunction of the Laplacian can take constant value, is the disc.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00942843
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