Positivity for polyharmonic problems on domains close to a disk

We study the problem of positivity preserving of the Green operator for the polyharmonic operator (?Δ) ^m under homogeneous Dirichlet boundary conditions on domains Ω of ?R ². Here we will treat only Ω, which are ε-close to a disk B in C ^m,γ-sense, meaning, there exists a C ^m,γ-mapping g : \( \bar{B}\longrightarrow \bar{\Omega}\) such that g?(B) = ?Ω and \(||g -- Id||_{C^{m,\gamma}}(\bar{B})\!\leq\!\varepsilon\). We show that ε-closeness in C ^{m, γ}-sense is enough in order to ensure positivity preserving. For the clamped plate equation (i.e. m = 2), this means that it is a Hölder norm of the curvature of ? Ω, which governs the positivity behavior. This improves the previous work by Grunau and Sweers, where closeness to the disk in C ^2m -sensewas required (in C ⁴-sense for thethe clamped plate).