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

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

Research output: Contribution to journalArticle

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 har-monicity 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.

Original languageEnglish
Pages (from-to)1205-1235
Number of pages31
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume39
Issue number3
DOIs
Publication statusPublished - 1 Mar 2019

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