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Stability of the positivity of biharmonic Green's functions under perturbations of the domain

by Grunau, H.-Ch., Robert, F.

Series: 2007-18, Preprints

35J40 Boundary value problems for higher-order elliptic equations
35B50 Maximum principles

In general, higher order elliptic equations and boundary value
problems like the biharmonic equation or the linear clamped plate
boundary value problem do not enjoy neither a maximum principle
nor a comparison principle or -- equivalently -- a positivity
preserving property. The problem is rather involved since the
clamped boundary conditions prevent the boundary value problem
{from} being written as a system of second order boundary value

On the other hand, 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.
Previously it was shown that this property also remains under small regular perturbations
of the domain, if $n=2$. In the present paper, such a stability result is proved for $n\ge 3$.

biharmonic Green's function, positivity, domain perturbations, dimension > 2