Weighted biharmonic Green functions for rational weights. (English) Zbl 0930.31003

This paper deals with weighted biharmonic operators of the form \(\Delta|P'|^2\Delta\) on the unit disc, where \(\Delta\) is the Laplacian and \(P\) is a rational function. The existence of the Green function for this operator is established for any rational function \(P\), and an algorithm is provided for obtaining it (by solving a certain system of linear equations). In the special case where \(P\) is a Blaschke product with two zeros, it is shown that the Green function is positive if and only if the hyperbolic distance between the two zeros does not exceed \((2/7)\sqrt{10}\). The author also obtains an explicit formula for the Green function of the operator \(\Delta|G|^{-2} \Delta\), where \(G\) is the canonical zero-divisor of a finite zero set on the Bergman space.


31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40 Boundary value problems for higher-order elliptic equations
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