# On nodal domains in Euclidean balls

@inproceedings{Helffer2015OnND,
title={On nodal domains in Euclidean balls},
author={Bernard Helffer and Mikael Persson Sundqvist},
year={2015}
}
• Published 12 June 2015
• Mathematics
A. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is ≥ 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality…

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*The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He
Let M be a closed n-dimensional manifold on which there is given a smooth positive density dx and let A be an elliptic, self-adjoint operator of degree m on M such that a m (x, ξ) > 0 for ξ ≠ 0. We