When is the first eigenfunction for the clamped plate equation of fixed sign?

Abstract

It is known that the first eigenfunction of the clamped plate equation, $Delta^2 varphi = lambda varphi$ in $Omega$ with $varphi=frac{partial}{partial n}varphi=0$ on $partialOmega$, is not necessarily of fixed sign. In this article, we survey the relations between domains $Omega$ and the sign of that first eigenfunction.

Keywords

• Clamped plate
• first eigenfunction
• maximum principle
• biharmonic.