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Weighted integrability of polyharmonic functions in the higher-dimensional case
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2021 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 14, no 7, p. 2047-2068Article in journal (Refereed) Published
Abstract [en]

This paper is concerned with the L-p integrability of N-harmonic functions with respect to the standard weights (1 - vertical bar x vertical bar(2))(alpha) on the unit ball B of R-n, n >= 2. More precisely, our goal is to determine the real (negative) parameters alpha for which (1 - vertical bar x vertical bar(2))(alpha/p) u(x) is an element of L-p(B) implies that u equivalent to 0 whenever u is a solution of the N-Laplace equation on B. This question is motivated by the uniqueness considerations of the Dirichlet problem for the N-Laplacian Delta(N).

Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464-505), where a complete answer to the above question in the case n D 2 is given for the full scale 0 < p < infinity. When n >= 3, we obtain an analogous characterization for n-2/n-1 <= p < infinity and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

Place, publisher, year, edition, pages
Mathematical Sciences Publishers (MSP) , 2021. Vol. 14, no 7, p. 2047-2068
Keywords [en]
polyharmonic functions, weighted integrability, boundary behavior, cellular decomposition
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-191680DOI: 10.2140/apde.2021.14.2047ISI: 000733976600002Scopus ID: 2-s2.0-85114153077OAI: oai:DiVA.org:umu-191680DiVA, id: diva2:1630784
Available from: 2022-01-21 Created: 2022-01-21 Last updated: 2023-03-24Bibliographically approved

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Perälä, Antti

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CiteExportLink to record
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Citation style
  • apa
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