  We set up a new framework to study critical points of functionals defined as
combinations of eigenvalues of operators with respect to a given set of
parameters: Riemannian metrics, potentials, etc. Our setting builds upon
Clarke's differentiation theory to provide a novel understanding of critical
metrics. In particular, we unify and refine previous research carried out on
Laplace and Steklov eigenvalues. We also use our theory to tackle original
examples such as the conformal GJMS operators, the conformal Laplacian, and the
Laplacian with mixed boundary conditions.
