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On nodal domains in Euclidean balls

Å. 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 $\geq 2$. 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 in Courant's theorem. In this note, we identify the Courant sharp eigenvalues for the Dirichlet and Neumann Laplacians in balls in $\mathbb R^d$, $d\geq 2$. The result for the Dirichlet Laplacian in the disc in $\mathbb R^2$ was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.