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
.
Keywords
maximum principle, fourth-order, weighted Dirichlet
bilaplace problem, positivity-preserving, positive
eigenfunction