Journal of the Mathematical Society of Japan

Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems


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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.

Article information

J. Math. Soc. Japan, Volume 68, Number 2 (2016), 773-796.

First available in Project Euclid: 15 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 32W20: Complex Monge-Ampère operators

Kiselman Monge–Ampère Kähler metrics


DARVAS, Tamás; RUBINSTEIN, Yanir A. Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems. J. Math. Soc. Japan 68 (2016), no. 2, 773--796. doi:10.2969/jmsj/06820773.

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