On 'anti'-eigenvalues for elliptic systems and a question of McKenna and Walter

We investigate for which range of b the biharmonic boundary value problem Delta(2)u + bu = f in Omega, with Deltau = u = 0 on partial derivativeOmega, is positivity-preserving in the sense that f greater than or equal to 0 in Omega implies u greater than or equal to 0. We will also disprove a conjecture of McKenna and Walter on the isoperimetric nature of the upper bound b(c) (Omega) for such b. The investigation gives rise to related questions for certain linear elliptic systems and to curious identities for sums of inverse eigenvalues.