Malmheden's theorem revisited

M. Agranovsky, D. Khavinson, H. S. Shapiro

Research output: Contribution to journalArticlepeer-review

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Abstract

In 1934 Malmheden [16] discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin (1957) [8] 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in Rn.

Original languageEnglish
Pages (from-to)337-350
Number of pages14
JournalExpositiones Mathematicae
Volume28
Issue number4
DOIs
StatePublished - 2010

Bibliographical note

Funding Information:
Some of this research was done as a part of European Science Foundation Networking Programme HCAA. The work of the first author was partially supported by the Israel Science Foundation under the Grant 688/08. The second author gratefully acknowledges partial support from the National Science Foundation Grant DMS-0855597. The authors are also indebted to the referee whose careful reading of the manuscript helped to improve the exposition.

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