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

TL;DR

This paper introduces new formulas linking convex analysis, Monge-Ampere equations, and Kiselman's principle, establishing partial regularity results and second-order regularity for rooftop obstacle problems, with applications to convex and plurisubharmonic envelopes.

Contribution

It provides a novel formula for Bremermann envelopes, relates Monge-Ampere solutions to Kiselman's principle, and proves second-order regularity for rooftop obstacle problems.

Findings

New formula for Bremermann envelopes via Legendre transform.

Partial regularity results for Bremermann envelopes.

Bounded second derivatives for rooftop obstacle envelopes.

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.