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Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems
- Source :
- J. Math. Soc. Japan 68 (2016), 773--796
- Publication Year :
- 2014
-
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.
- Subjects :
- Mathematics - Analysis of PDEs
Mathematics - Complex Variables
Subjects
Details
- Database :
- arXiv
- Journal :
- J. Math. Soc. Japan 68 (2016), 773--796
- Publication Type :
- Report
- Accession number :
- edsarx.1405.6548
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2969/jmsj/06820773