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

Fingerprint

Decay of Solutions
Half-space
Dirichlet Boundary Conditions
Boundary value problems
Dirichlet
Explicit Formula
Inversion
Ball
Uniqueness
Regularity
Boundary Value Problem
Infinity
Boundary conditions
Operator
Estimate

Cite this

Abatangelo, Nicola ; Dipierro, Serena ; Fall, Mouhamed Moustapha ; Jarohs, Sven ; Saldaña, Alberto. / Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. In: Discrete and Continuous Dynamical Systems- Series A. 2019 ; Vol. 39, No. 3. pp. 1205-1235.
@article{3d9c3ab1f4874c7c9d810958d8233853,
title = "Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions",
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.",
keywords = "Fractional Laplacian, Green function, Kelvin transform, Nonlocal operator, Poisson kernel, S-harmonicity",
author = "Nicola Abatangelo and Serena Dipierro and Fall, {Mouhamed Moustapha} and Sven Jarohs and Alberto Salda{\~n}a",
year = "2019",
month = "3",
day = "1",
doi = "10.3934/dcds.2019052",
language = "English",
volume = "39",
pages = "1205--1235",
journal = "DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS. SERIES A",
issn = "1078-0947",
publisher = "Southwest Missouri State University",
number = "3",

}

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. / Abatangelo, Nicola; Dipierro, Serena; Fall, Mouhamed Moustapha; Jarohs, Sven; Saldaña, Alberto.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 39, No. 3, 01.03.2019, p. 1205-1235.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

AU - Abatangelo, Nicola

AU - Dipierro, Serena

AU - Fall, Mouhamed Moustapha

AU - Jarohs, Sven

AU - Saldaña, Alberto

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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.

AB - 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.

KW - Fractional Laplacian

KW - Green function

KW - Kelvin transform

KW - Nonlocal operator

KW - Poisson kernel

KW - S-harmonicity

UR - http://www.scopus.com/inward/record.url?scp=85058935983&partnerID=8YFLogxK

U2 - 10.3934/dcds.2019052

DO - 10.3934/dcds.2019052

M3 - Article

VL - 39

SP - 1205

EP - 1235

JO - DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS. SERIES A

JF - DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS. SERIES A

SN - 1078-0947

IS - 3

ER -