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First, we obtain a new formula for Bremermann type upper envelopes, thatarise frequently in convex analysis and pluripotential theory, in terms of theLegendre transform of the convex- or plurisubharmonic-envelope of the boundarydata. This yields a new relation between solutions of the Dirichlet problem forthe homogeneous real and complex Monge-Ampere equations and Kiselman's minimumprinciple. More generally, it establishes partial regularity for a Bremermannenvelope whether or not it solves the Monge-Ampere equation. Second, we provethe second order regularity of the solution of the free-boundary problem forthe Laplace equation with a rooftop obstacle, based on a new a priori estimateon the size of balls that lie above the non-contact set. As an application, weprove that convex- and plurisubharmonic-envelopes of rooftop obstacles havebounded second derivatives.

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