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We study the problem of finding the instability index of certainnon-selfadjoint fourth order differential operators that appear aslinearizations of coating and rimming flows, where a thin layer of fluid coatsa horizontal rotating cylinder. The main result reduces the computation of theinstability index to a finite-dimensional space of trigonometric polynomials.The proof uses Lyapunov's method to associate the differential operator with aquadratic form, whose maximal positive subspace has dimension equal to theinstability index. The quadratic form is given by a solution of Lyapunov'sequation, which here takes the form of a fourth order linear PDE in twovariables. Elliptic estimates for the solution of this PDE play a key role. Weinclude some numerical examples.

On computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows