A representation formula for radially weighted biharmonic functions in the unit disc

Abstract

Let $w\colon \mathbb{D}\to(0,\infty)$ be a radial continuous weight function. We consider the weighted biharmonic equation \Delta w^{-1}\Delta u=0\quad\text{in } \mathbb{D} with Dirichlet boundary conditions $u = f_0$ and $\partial_n u = f_1$ on $\mathbb{T} = \partial\mathbb{D}$. Under some extra conditions on the weight function $w$, we establish existence and uniqueness of a distributional solution $u$ of this biharmonic Dirichlet problem. Furthermore, we give a representation formula for the solution $u$. The key to our analysis is a series representation of Almansi type.