Differential tests for plurisubharmonic functions and Koch curves

2016/07/04 by Dinew, Sławomir, Dinew, Żywomir
#28A78 #28A80 #30C62 #31C10 #32U05 #32W20 #35D40 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.00893

Abstract

We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper tests with a control on the opening.

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