Abstract
We study the minimum sets of plurisubharmonic functions with strictly positive Monge–Ampere densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.
Highlights
A classical theorem of Harvey and Wells [21] states that the zero set of a nonnegative strictly plurisubharmonic and smooth function is contained in a C1 totally real submanifold
One of them is that compact pieces of such satisfy the Condition (P) introduced by Catlin in [12] which is crucial for the compactness of the ∂-Neumann problem
In a completely different direction El Mir [17] has shown that zero sets of bounded continuous strictly plurisubharmonic functions are removable sets in the theory of extensions of closed positive currents
Summary
A classical theorem of Harvey and Wells [21] states that the zero set of a nonnegative strictly plurisubharmonic and smooth function is contained in a C1 totally real submanifold. On the other hand Blank himself gave a very interesting example in [2] showing that for f less than Dini smooth the free boundary can be badly behaved- in particular it can spin around a point infinitely often All this suggested that the minimum sets, just like free boundaries, can be badly behaved but are of Hausdorff dimension less or equal to one. Our result disproves that: Theorem 4 In the planar case there are compact sets K and F B, such that K is a minimum set of a strictly subharmonic function and F B is a free boundary such that dimH K = dimH F B > 1. To our knowledge the free boundary problem for the complex Monge–Ampère equation has not been thoroughly investigated and we plan to consider this in a future article [14]
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