Our main result is that for any bounded smooth domain there exists a positive-weight function and an interval such that for and in with on the following holds: if is positive, then is positive. The proofs are based on the construction of an appropriate weight function with a corresponding strongly positive eigenfunction and on a converse of the Krein–Rutman theorem. For the Dirichlet bilaplace problem above with 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 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 .
"A maximum principle for a fourth-order Dirichlet problem on smooth domains." Pure Appl. Anal. 2 (3) 685 - 702, 2020. https://doi.org/10.2140/paa.2020.2.685