论文标题
具有不连续边界行为的Plurisubharmonic函数
Plurisubharmonic functions with discontinuous boundary behavior
论文作者
论文摘要
我们研究了具有有界的，不连续的边界数据的复杂Monge-Ampère操作员的DIRICHLET问题。如果一组不连续性是B-倍极，并且该域是b的，那么我们能够证明一系列复杂的Monge-Ampère措施的存在，独特性和一些规律性估计。该结果在单位磁盘中是最佳的，因为边界函数具有B-倍极不连续性，然后与几乎无处不在的函数一致。我们还表明，边界函数的这些属性都不是连续的，几乎无处不在或形成B-倍极集的不连续性 - 是为了在更高维度中建立溶液的唯一性和连续性。特别是，在某些情况下，在一组任意小的Lebesgue度量下开出边界行为就足以规定边界行为。
We study the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex Monge-Ampère measures. This result is optimal in the unit disk, as boundary functions with b-pluripolar discontinuity then coincides with functions that are continuous almost everywhere. We also show that neither of these properties of the boundary function - being continuous almost everywhere or having discontinuities forming a b-pluripolar set - are necessary conditions in order to establish uniqueness and continuity of the solution in higher dimensions. In particular, there are situations where it is enough to prescribe the boundary behavior at a set of arbitrarily small Lebesgue measure.
