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May  2018, 17(3): 899-922. doi: 10.3934/cpaa.2018045

## Positive powers of the Laplacian: From hypersingular integrals to boundary value problems

 1 Département de Mathématique, Université Libre de Bruxelles, boulevard du Triomphe CP 214, 1050 Bruxelles, Belgium 2 Institut für Mathematik, Goethe-Universität, Robert-Mayer-Straße 10, 60054 Frankfurt, Germany 3 Institut für Analysis, Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany

Received  September 2017 Revised  September 2017 Published  January 2018

Fund Project: The third author was supported by a research fellowship from the Alexander von Humboldt Foundation.

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.

Citation: Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian: From hypersingular integrals to boundary value problems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 899-922. doi: 10.3934/cpaa.2018045
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