### 2007-18

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

**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