Weak solutions to the quaternionic Monge–Ampère equation

Sroka, Marcin

doi:10.2140/apde.2020.13.1755

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Weak solutions to the quaternionic Monge–Ampère equation

Marcin Sroka

Vol. 13 (2020), No. 6, 1755–1776

DOI: 10.2140/apde.2020.13.1755

Abstract

We solve the Dirichlet problem for the quaternionic Monge–Ampère equation with a continuous boundary data and the right-hand side in $L^{p}$ for $p > 2$ . This is the optimal bound on $p$ . We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is 2, which turns out to be less than an integrability exponent of the fundamental solution.

Keywords

Monge–Ampère equation, pluripotential theory, quaternionic plurisubharmonic functions

Mathematical Subject Classification 2010

Primary: 32U05, 32U15, 35D30, 35J60

Milestones

Received: 19 July 2018

Revised: 9 May 2019

Accepted: 11 July 2019

Published: 12 September 2020

Authors

Marcin Sroka
Faculty of Mathematics and Computer Science Jagiellonian University Kraków Poland

Vol. 13, No. 6, 2020