A computation of Poisson kernels for some standard weighted biharmonic operators in the unit disc

Abstract

Abstract. 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. IntroductionWe address in this paper the problem of finding explicit formulas for Poisson kernels for weighted biharmonic operators of the form ∆w −1 ∆ in the unit disc D with Dirichlet boundary conditions, where ∆ = ∂ 2 /∂z∂z, z = x + iy, is the Laplacian in the complex plane and w = w γ is a weight function of the formfor some real parameter γ > −1. Such a weight function w γ 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 problemHere T = ∂D is the unit circle and ∂ n denotes differentiation in the inward normal direction. The first equation in (0.1), the biharmonic equation ∆w −1 ∆u = 0, is evaluated in the distributional sense and defines a class of functions which we call w-biharmonic. In full generality the boundary datas f j ∈ D ′ (T) (j = 0, 1) are distributions on T and the boundary conditions in (0.1) are interpreted in a distributional sense as follows: Let u be a smooth function in D and let f 0 ∈ D ′ (T). We say that u = f 0 on T in the distributional sense if lim r→1 u r = f 0 in D ′ (T), where (0.2) u r (e iθ ) = u(re iθ ), e iθ ∈ T,

Weighted integrability of polyharmonic functions

“…the Laplacian ∆ is associated with a membrane, the bilaplacian ∆ 2 is associated with a plate. There is a sizeable literature related to the bilaplacian, and more generally the N-laplacian ∆ N ; see, e.g., the books [4], [10], [12], [14], and the papers [11], [20], [21], [23], [1], [15], [32], [33]. We could also mention that the biharmonic Green function was used in [8] to generalize the factorization of L 2 Bergman space functions found in [18] to the L p setting (see also [19], [3]).…”

“…We also note that the function U 2,2 appears implicitly in the biharmonic setting in [1] (U 2,2 (z) = 2F(z, 1) − H(z, 1) in their notation); cf. [33]. More recently, in [34] the kernels U N,N are shown to solve the Dirichlet problem in the disk D for a certain (singular) second order elliptic differential operator.…”

“…There is a sizeable literature related to the bilaplacian, and more generally the N-laplacian ∆ N ; see, e.g., the books [4], [10], [12], [14], and the papers [11], [20], [21], [23], [1], [15], [32], [33].…”