Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime

Guido Sweers

doi:10.3934/dcdss.2014.7.839

Article Contents

2014, Volume 7, Issue 4: 839-855. Doi: 10.3934/dcdss.2014.7.839

This issue Previous Article Hopf fibration and singularly perturbed elliptic equations Next Article Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators

Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime

Guido Sweers^1,

1.
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln

Received: July 31, 2013

Revised: October 31, 2013

Published: July 2014

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Abstract

We address the question, for which $\lambda \in \mathbb{R}$ is the boundary value problem \begin{equation*} \left\{ \begin{array}{cc} \Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\ u=\Delta u=0 & \text{on }\partial \Omega , \end{array} \right. \end{equation*} positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what happens, when $\lambda $ passes the maximal value for which positivity is preserved.

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Mathematics Subject Classification: Primary: 35J40; Secondary: 35B50, 60J85.

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