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2020 A maximum principle for a fourth-order Dirichlet problem on smooth domains
Inka Schnieders, Guido Sweers
Pure Appl. Anal. 2(3): 685-702 (2020). DOI: 10.2140/paa.2020.2.685

Abstract

Our main result is that for any bounded smooth domain Ωn there exists a positive-weight function w and an interval I such that for λI 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.

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Inka Schnieders. Guido Sweers. "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

Information

Received: 13 January 2020; Revised: 7 May 2020; Accepted: 3 July 2020; Published: 2020
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.2140/paa.2020.2.685

Subjects:
Primary: 35B50
Secondary: 35J40, 47B65

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.2 • No. 3 • 2020
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