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Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. (English) Zbl 0838.35006
Summary: We describe the boundary behavior of the nodal lines of eigenfunctions of the fixed membrane problem in convex, possibly nonsmooth, domains. This result is applied to the proof of Payne’s conjecture on the nodal line of second eigenfunctions, by removing the \(C^\infty\) smoothness assumption which is present in the original proof of A. D. Melas [J. Differ. Geom. 35, No. 1, 255-263 (1992; Zbl 0769.58056)].

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J67 Boundary values of solutions to elliptic equations and elliptic systems
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