Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type

Abstract

Abstract. Let Ω be a bounded, pseudoconvex domain of C n satisfying the "f -Property". The f -Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equationThe idea of our proof goes back to Bedford and Taylor's [BT76]. However, the basic geometrical ingredient is based on a recent result by Khanh [Kha13].

Hölder regularity of the solution to the complex Monge-Ampère equation with $$L^p$$ L p density

“…Ω). Ha and Khanh in [9] get the same conclusion with a more geometric notion of finite type (cf. (1.2) below) and have also a generalization for the infinite type.…”

“…First, from Khanh and Zampieri [11], we know that (1.2) implies the potential-theoretic "t 1 m -property". By [10] and [9] this implies in turn that there is an exhaustion function ρ which defines Ω by ρ < 0 such that Then the unique solution u to MA(Ω, ϕ, f ) is in C min( α m , γ m ) ( Ω) with γ < γ p where γ p := 2 qn+1 and 1 p + 1 q =1. The proof follows in Section 3.…”

“…First, from Khanh and Zampieri [11], we know that (1.2) implies the potential-theoretic "t 1 m -property". By [10] and [9] this implies in turn that there is an exhaustion function ρ which defines Ω by ρ < 0 such that…”

“…Proof. We consider the solution u(Ω, ϕ, 0) ∈ C α m (Ω) by [18] and [9] and define v = u(Ω, ϕ, 0) + cρ.…”

A geometric approach to Catlin’s boundary systems

“…(1) Compactness of the Kohn's∂-Neumann solution operator by Henkin [HK15] and Baracco, Khanh, Pinton and Zampieri [BKPZ16]. For bounded pseudoconvex domains with real-analytic boundaries of finite type in C n , Kohn's [K79] celebrated theory of subelliptic multipliers provides an alternative approach to Catlin's in establishing subelliptic estimates.…”