A biharmonic converse to Krein–Rutman: a maximum principle near a positive eigenfunction

Abstract

The Green function \(G_0(x,y)\) for the biharmonic Dirichlet problem on a smooth domain \(\Omega \), that is \(\Delta ^{2}u=f\) in \(\Omega \) with \( u=u_{n}=0 \) on \(\partial \Omega \), 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 \(G_0(x,y)= H(x,y)- c\, d(x)^2 d(y)^2\) holds, where \(d(\cdot )\) is the distance to the boundary \(\partial \Omega \) and where H contains the singularity and is positive. We will extend the corresponding estimates to \( G_{\lambda }(x,y)\) for the differential operator \(\Delta ^{2}-\lambda \) with an optimal dependence on \(\lambda \). As a consequence, strict positivity of an eigenfunction with a simple eigenvalue \(\lambda _{i}\) implies a positivity preserving property for \(\left( \Delta ^{2}-\lambda \right) u=f\) in \(\Omega \) with \(u=u_{n}=0\) on \(\partial \Omega \) for \(\lambda \) in a left neighbourhood of \(\lambda _{i} \). This result can be viewed as a converse to the Krein–Rutman theorem.