Positivity for polyharmonic problems on domains close to a disk

摘要

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 ℝR 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).