Positivity for the clamped plate equation under high tension

TL;DR

This paper investigates positivity properties of solutions to the high-tension clamped plate equation, establishing conditions under which the solution remains positive for sufficiently large tension parameter, applicable in all dimensions.

Contribution

It provides a dimension-independent approach to ensure positivity of solutions for high-tension clamped plate equations, extending previous results.

Findings

Existence of a tension threshold  depending on domain and source norms

Positivity of the solution for all  with sufficiently large tension

Applicable in all spatial dimensions

Abstract

In this article we consider positivity issues for the clamped plate equation with high tension $\gamma>0$. This equation is given by $\Delta^2u - \gamma\Delta u=f$ under clamped boundary conditions. Here we show, that given a positive $f$, i.e. upwards pushing, we find a $\gamma_0>0$ such that for all $\gamma\geq \gamma_0$ the bending $u$ is indeed positive. This $\gamma_0$ only depends on the domain and the ratio of the $L^1$ and $L^\infty$ norm of $f$. In contrast to a recent result by Cassani&Tarsia, our approach is valid in all dimensions.