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

Abstract

Our main result is that for any bounded smooth domain $\mathrm{\Omega }\subset {ℝ}^n$ there exists a positive-weight function $w$ and an interval $I$ such that for $\lambda \in I$ and ${\mathrm{\Delta }}^{2}u=\lambda wu+f$ in $\mathrm{\Omega }$ with $u=\frac{\partial }{\partial \nu }u=0$ on $\partial \mathrm{\Omega }$ 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 $\lambda =0$ 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 $\lambda$.