A Computation of Poisson Kernels for Some Standard Weighted Biharmonic Operators in the Unit Disc
نویسنده
چکیده
We compute Poisson kernels for integer weight parameter standard weighted biharmonic operators in the unit disc with Dirichlet boundary conditions. The computations performed extend the supply of explicit examples of such kernels and suggest similar formulas for these Poisson kernels to hold true in more generality. Computations have been carried out using the open source computer algebra package Maxima. 0. Introduction We address in this paper the problem of finding explicit formulas for Poisson kernels for weighted biharmonic operators of the form ∆w∆ in the unit disc D with Dirichlet boundary conditions, where ∆ = ∂/∂z∂z̄, z = x + iy, is the Laplacian in the complex plane and w = wγ is a weight function of the form wγ(z) = (1− |z| ) , z ∈ D, for some real parameter γ > −1. Such a weight function wγ is commonly referred to as a standard weight. Let us first describe the context of these Poisson kernels. Let w : D → (0,∞) be a smooth radial weight function and consider the weighted biharmonic Dirichlet problem
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