Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

Nicola Abatangelo; Serena Dipierro; Mouhamed Moustapha Fall; Sven Jarohs; Alberto Saldaña

doi:10.3934/dcds.2019052

Article Contents

2019, Volume 39, Issue 3: 1205-1235. Doi: 10.3934/dcds.2019052

This issue Previous Article Preface Next Article Fundamental solutions of a class of homogeneous integro-differential elliptic equations

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

1.
Département de mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, 1050 Ixelles, Belgium
2.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia
3.
African Institute for Mathematical Sciences (A.I.M.S) of Senegal, KM 2, Route de Joal (Centre I.R.D. Mbour), B.P. 1418 Mbour, Sénégal
4.
Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Straße 10, 60054 Frankfurt, Germany
5.
Institut für Analysis, Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany

* Corresponding author: Sven Jarohs
* Corresponding author: Sven Jarohs

Received: May 31, 2018

Revised: July 31, 2018

Published: December 2018

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Abstract

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

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Mathematics Subject Classification: Primary: 35C05, 35C15, 35S15; Secondary: 35B50.

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