Abstract
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).
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Mathematics Subject Classification (2000) 35J30, 35B50
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Sassone, E. Positivity for polyharmonic problems on domains close to a disk. Annali di Matematica 186, 419–432 (2007). https://doi.org/10.1007/s10231-006-0012-3
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DOI: https://doi.org/10.1007/s10231-006-0012-3