In any dimension a “clamped plate” with a uniform weight may change sign

Positivity preserving properties have been conjectured for the bilaplace Dirichlet problem in many versions. In this note we show that in any dimension there exist bounded smooth domains ΩΩ such that even the solution of Δ2u=1Δ2u=1 in ΩΩ with the homogeneous Dirichlet boundary conditions u=uν=0u=uν=0 on ∂Ω∂Ω is sign-changing. In two dimensions this corresponds to the Kirchhoff–Love model of a clamped plate with a uniform weight.