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Article overview
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Weighted integrability of polyharmonic functions | | Haakan Hedenmalm
; Alexander Borichev
; | | Date: |
21 Nov 2012 | | 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
$pmapstoeta(N,p)$ -- a piecewise affine function -- such that for
$alpha>eta(N,p)$ there exist non-trivial functions $u$ with $Delta^N u=0$
of the given integrability, while for $alphaleeta(N,p)$, 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. | | Source: | arXiv, 1211.5088 | | Services: | Forum | Review | PDF | Favorites | |
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