The equation of complex Monge-Ampère type and stability of solutions

Abstract

We prove that in a family of plurisubharmonic functions with Monge-Ampère measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampère type equation. Subject Classification (1991): 32U05, 32U40 Mathematics

Convergence in Capacity

“…Therefore, Stability Theorem follows from Corollary 2.12. Moreover, since Corollary 2.12 is valid for all type of bounded domains, it also implies the stability theorem for hyperconvex domains, see [10].…”

“…-We do not assume in Corollary 2.12 that all the functions u j have the same continuous boundary data. In fact, under the assumptions of the stability theorem in [10] there exist functions…”

“…Denote by F(φ, Ω) the class of those u ∈ P SH(Ω) for which h u h + v for some v ∈ F(Ω). We use the subclass F a (φ, Ω) of functions from F(φ, Ω) whose Monge-Ampère measures put no mass on all pluripolar subsets of Ω, see [1] [10]. Clearly, F(Ω) = F(0, Ω) and F a (Ω) = F a (0, Ω).…”

Monge–Ampère boundary measures

“…From Lemma 1.9 in [13], using the fact that u ∈ F a (Ω) and cap(U ε , W ) < ε, it follows that this last integral tends to zero as ε 0, which completes the proof.…”

On the Cegrell classes

“…Lemma 5.1 was proved in [9] for functions from F . As noted in [13] the proof carries over to E. Therefore it is also valid for functions from E(f ). Lemma 5.1 Let f : ∂ → R be a continuous function such that (2.1) holds.…”