Positivity (2020) 24:677–710
https://doi.org/10.1007/s11117-019-00702-3
Positivity
A biharmonic converse to Krein–Rutman: a maximum
principle near a positive eigenfunction
Inka Schnieders
1
· Guido Sweers
1
Received: 27 August 2018 / Accepted: 17 August 2019 / Published online: 23 August 2019
© Springer Nature Switzerland AG 2019
Abstract
The Green function G
0
(x , y ) for the biharmonic Dirichlet problem on a smooth domain
, that is
2
u = f in with u = u
n
= 0 on ∂, can be written as the difference
of a positive function, which bears the singularity at x = y , and a rank-one positive
function, both of which satisfy the boundary conditions. See Grunau et al. (Proc Am
Math Soc 139:2151–2161, 2011). More precisely G
0
(x , y ) = H (x , y ) −cd (x )
2
d ( y )
2
holds, where d (·) is the distance to the boundary ∂ and where H contains the sin-
gularity and is positive. We will extend the corresponding estimates to G
λ
(x , y ) for
the differential operator
2
− λ with an optimal dependence on λ. As a consequence,
strict positivity of an eigenfunction with a simple eigenvalue λ
i
implies a positivity
preserving property for
(
2
− λ
)
u = f in with u = u
n
= 0 on ∂ for λ in a left
neighbourhood of λ
i
. This result can be viewed as a converse to the Krein–Rutman
theorem.
Keywords Maximum principle · Biharmonic Dirichlet problem · Positivity
preserving · Positive eigenfunction
Mathematics Subject Classification Primary 35B50; Secondary 35J40 · 47B65
1 Introduction
For second order elliptic boundary value problems the standard maximum principle
yields a very useful ordering result, which is also called maximum principle. Indeed,
for Lu = f on a domain ⊂ R
n
with for example L =− , the assumption f ≥ 0
B Guido Sweers
gsweers@math.uni-koeln.de
Inka Schnieders
ischnied@math.uni-koeln.de
1
Department of Mathematics and Computer Science, Universität zu Köln, Weyertal 86-90,
50931 Cologne, Germany
123