Anti-eigenvalues and a conjecture of McKenna and Walter

Bernhard Kawohl Univ. of Cologne, Cologne, Germany

McKenna and Walter remarked in 1987 that the following linear problem needs more study:

(1)  \{ \begin{matrix}Δ^{2}v + b v & = & f & in Ω, \\ v = Δv & = & 0 & on ∂Ω, \end{matrix}

where Ω is a bounded domain in ℝ^{n} and f a given function:

… we should be able to estimate the value b_{c}(Ω) with the property that for 0 ≤ b < b_{c}(Ω) the inverse operator is positivity preserving (…). We conjecture that this constant b_{c}(Ω) is largest among all regions Ω with given volume when Ω is a ball.

In the lecture I present results on this conjecture and on related questions that were obtained jointly with G.Sweers from Delft. The parameter b can be interpreted as a spring constant, and u as a deformation of a hinged plate in a bed of springs under a load f.