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April, 2016 Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems
Tamás DARVAS, Yanir A. RUBINSTEIN
J. Math. Soc. Japan 68(2): 773-796 (April, 2016). DOI: 10.2969/jmsj/06820773

Abstract

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge–Ampère equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge–Ampère equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.

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Tamás DARVAS. Yanir A. RUBINSTEIN. "Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems." J. Math. Soc. Japan 68 (2) 773 - 796, April, 2016. https://doi.org/10.2969/jmsj/06820773

Information

Published: April, 2016
First available in Project Euclid: 15 April 2016

zbMATH: 1353.32039
MathSciNet: MR3488145
Digital Object Identifier: 10.2969/jmsj/06820773

Subjects:
Primary: 53C55
Secondary: 32W20

Rights: Copyright © 2016 Mathematical Society of Japan

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Vol.68 • No. 2 • April, 2016
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