Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

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

Pages: 839 - 855, Volume 7, Issue 4, August 2014      doi:10.3934/dcdss.2014.7.839

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Guido Sweers - Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany (email)

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.

Keywords:  Bilaplace, Navier boundary, positivity, anti-eigenvalue, expected lifetime, Green function.
Mathematics Subject Classification:  Primary: 35J40; Secondary: 35B50, 60J85.

Received: August 2013;      Revised: November 2013;      Available Online: February 2014.