## 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.

### MSC:

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

### Keywords:

biharmonic function; Green function; Bergman space
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