The clamped-plate equation for the limaçon

Abstract

Hadamard claimed in 1907 that the clamped-plate equation is positivity preserving for domains which are bounded by a Limaçon de Pascal. We will show that this claim is false in its full generality. However, we will also prove that there are nonconvex limaçons for which the clamped-plate equation has the sign-preserving property. In fact we will give an explicit bound for the parameter of the limaçon where sign change may occur.

Keywords clamped plate - biharmonic - positivity - nonconvex

Mathematics Subject Classification (2000) 35J40, 35Q72, 35B50