Weighted integrability of polyharmonic functions

Abstract

To address the uniqueness issues associated with the Dirichlet problem for the $N$-harmonic equation on the unit disk $\D$ in the plane, we investigate the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|z|^2)^{\alpha}$. The question at hand is the following. If $u$ solves $\Delta^N u=0$ in $\D$, where $\Delta$ stands for the Laplacian, and [\int_\D|u(z)|^p (1-|z|^2)^{\alpha}\diff A(z)<+\infty,] must then $u(z)\equiv0$? Here, $N$ is a positive integer, $\alpha$ is real, and $0<p<+\infty$; $\diff A$ is the usual area element. The answer will, generally speaking, depend on the triple $(N,p,\alpha)$. The most interesting case is $0<p<1$. For a given $N$, we find an explicit critical curve $p\mapsto\beta(N,p)$ -- a piecewise affine function -- such that for $\alpha>\beta(N,p)$ there exist non-trivial functions $u$ with $\Delta^N u=0$ of the given integrability, while for $\alpha\le\beta(N,p)$, only $u(z)\equiv0$ is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in $\mathrm{PH}^p_{N,\alpha}(\D)$ when this space is nontrivial. We find a fascinating structural decomposition of the polyharmonic functions -- the cellular (Almansi) expansion -- which decomposes the polyharmonic weighted $L^p$ in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the $(p,\alpha)$ plane into cells. A particularly interesting collection of cells form the entangled region.