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

Abstract

On a smooth domain Ω ⊂⊂ C n , we consider the Dirichlet problem for the complex Monge-Ampère equation ((dd c u) n = f dV, u| bΩ ≡ ϕ). We state the Hölder regularity of the solution u when the boundary value ϕ is Hölder continuous and the density f is only

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

“…Using this characterisation (Theorem 2.5) we can reprove the results from [2], [8], [15] in the case of zero boundary. We also show in Corollary 2.13 that there are several class of measures which satisfy the assumptions of the theorem.…”

“…Finally, the extra assumptions were removed in [2], [8]. However, there is still an open question to find a characterisation for the measures admitting Hölder continuous solutions to the equation.…”

Hölder continuous solutions to the complex Monge–Ampère equations in non-smooth pseudoconvex domains

“…Proof We use the technique of Li [30] (also see [2]). By the hypotheses it implies that h ξ ∈ P SH(Ω), ∀ξ ∈ ∂Ω.…”

“…Charabati [13] proved the Hölder regularity for solutions to M A(Ω, φ, f ) in bounded strongly hyperconvex Lipschitz domain. Recently, Baracco, Khanh, Pinton and Zampieri [2] generalized the theorem of [17] to C 2 smooth bounded pseudoconvex domain of plurisubharmonic type m under the assumption that the boundary data φ ∈ C α (∂Ω) with 0 < α ≤ 2. Note that the technique of [2] is not valid when Ω is not C 2 smooth.…”

“…Main purpose of this paper is to generalize the theorem of [2] from C 2 smooth bounded pseudoconvex domain of plurisubharmonic type m to nonsmooth pseudoconvex domains of plurisubharmonic type m. First we give the following definition which is an extension of Li [30] (also see [2]). Definition 1 Let m > 0 and let Ω be a pseudoconvex domain in C n .…”