  \AA. 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.

Spectral Theory