Stability of the positivity of biharmonic Green's functions under perturbations of the domain
Hans-Christoph Grunau, Frédéric Robert · 2012
Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday 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 ⊂ 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 ≥ 3. 1.