Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems

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-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere 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.

References in corpus (3)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• Kahler-Einstein metrics with edge singularities
• The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

Cited by in corpus (3)

• On the singularity type of full mass currents in big cohomology classes
• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• Geometry and Topology of the space of Kähler metrics on singular varieties