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.
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