Vol. 2, No. 3, 2020

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

Inka Schnieders and Guido Sweers

Vol. 2 (2020), No. 3, 685–702
Abstract

Our main result is that for any bounded smooth domain Ω n 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.

Keywords
maximum principle, fourth-order, weighted Dirichlet bilaplace problem, positivity-preserving, positive eigenfunction
Mathematical Subject Classification 2010
Primary: 35B50
Secondary: 35J40, 47B65
Milestones
Received: 13 January 2020
Revised: 7 May 2020
Accepted: 3 July 2020
Published: 17 November 2020
Authors
Inka Schnieders
Department Mathematik/Informatik
Universität zu Köln
Köln
Germany
Guido Sweers
Department Mathematik/Informatik
Universität zu Köln
Köln
Germany