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 Lp integrability of N-harmonic functions with respect to the standard weights (1−|z|2)α. The question at hand is the following. If u solves ΔNu=0 in D, where Δ stands for the Laplacian, and∫D|u(z)|p(1−|z|2)αdA(z)<+∞, must then u(z)≡0? Here, N is a positive integer, α is real, and 0<p<+∞; dA is the usual area element. The answer will, generally speaking, depend on the triple (N,p,α). The most interesting case is 0<p<1. For a given N, we find an explicit critical curve p↦β(N,p) – a piecewise affine function – such that for α>β(N,p) there exist nontrivial functions u with ΔNu=0 of the given integrability, while for α≤β(N,p), only u(z)≡0 is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in PHN,αp(D) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions – the cellular decomposition – which decomposes the polyharmonic weighted Lp space in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the (p,α) plane into cells.The above uniqueness for the Dirichlet problem may be considered for any elliptic operator of order 2N. However, the above-mentioned critical integrability curve will depend rather strongly on the given elliptic operator, even in the constant coefficient case, for N>1.