  The oscillation of a Laplacian eigenfunction gives a great deal of
information about the manifold on which it is defined. This oscillation can be
encoded in the nodal deficiency, an important geometric quantity that is
notoriously hard to compute, or even estimate. Here we compare two recently
obtained formulas for the nodal deficiency, one in terms of an energy
functional on the space of equipartitions of the manifold, and the other in
terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We
relate these two approaches by giving an explicit formula for the Hessian of
the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows
us to compute Hessian eigenfunctions, and hence directions of steepest descent,
for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann
eigenfunctions. Our results do not assume bipartiteness, and hence are relevant
to the study of spectral minimal partitions.
