A maximum principle for a fourth-order Dirichlet problem on smooth domains

Abstract

Our main result is that for any bounded smooth domain Ω⊂ℝn there exists a positive-weight function w and an interval I such that for λ∈I and Δ2u=λwu+f in Ω with u=∂∂νu=0 on ∂Ω the following holds: if f is positive, then u is positive. The proofs are based on the construction of an appropriate weight function w with a corresponding strongly positive eigenfunction and on a converse of the Krein–Rutman theorem. For the Dirichlet bilaplace problem above with λ=0 the Boggio–Hadamard conjecture from around 1908 claimed that positivity is preserved on convex 2-dimensional domains and was disproved by counterexamples from Duffin and Garabedian some 40 years later. With w=1 not even the first eigenfunction is in general positive. So by adding a certain weight function our result shows a striking difference: not only is a corresponding eigenfunction positive but also a fourth-order “maximum principle” holds for some range of λ.