The Generalized Gradient at a Multiple Eigenvalue

When a symmetric, positive, isomorphism between a reflexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an arbitrary crossing. We apply this to the generalized gradient, with respect to a coefficient in an elliptic operator, of(i) the gap between the operator′s first two eigenvalues and (ii) the distance from a prescribed value to the spectrum of the operator.