Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions

TL;DR

This paper investigates the positivity properties of biharmonic Green's functions under Dirichlet boundary conditions, demonstrating that the negative part is small in higher dimensions and establishing stability of positivity under domain perturbations.

Contribution

It proves that in dimensions three and higher, the negative part of biharmonic Green's functions is small and that positivity is stable under domain perturbations.

Findings

Negative part of Green's function is small in dimensions n≥3

Explicit positivity of Green's function in balls under Dirichlet conditions

Positivity stability under domain perturbations for n≥3

Abstract

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains $\Omega \subset\mathbb{R}^n$, the negative part of the corresponding Green's function is "small" when compared with its singular positive part, provided $n\ge 3$. Moreover, the biharmonic Green's function in balls $B\subset\mathbb{R}^n$ under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if $n=2$. In the present paper, such a stability result is proved for $n\ge 3$. Keywords: Biharmonic Green's functions, positivity, almost positivity, blow-up procedure.