2007-18

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

Series: 2007-18, Preprints

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

Abstract:
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
problems.

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\$.

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