Weighted integrability of polyharmonic functions

By A. Borichev, H. Hedenmalm. 2012

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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 ) α . The question at hand is the following. If u solves Δ N u = 0 in D , where Δ stands for the Laplacian, and ∫ D | u ( z ) | p ( 1 − | z | 2 ) α d A ( z ) + ∞ , must then u ( z ) ≡ 0 ? Here, N is a positive integer, α is real, and 0 p + ∞ ; d A 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 Δ N u = 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 PH N , α p ( D ) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions – the cellular decomposition – which decomposes the polyharmonic weighted L p 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 .

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/24773dbf8f1728da8dc83f5fdb1806f4245bf05c

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