Hölder continuous solutions to Monge-Ampère equations

Abstract

We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain Ω⊂ℂn. We show, under a mild technical assumption, that the unique solution u to this problem is Hölder continuous if the boundary data ϕ is Hölder continuous and the density f belongs to Lp(Ω) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.

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

“…Proof The proof is almost the same as the one given by [17]. For convenience to readers, we sketch the proof of the lemma.…”

“…Since q(1 + nτ ) > 1, by Proposition 1.4 in [17] there exists a constant C τ > 0 independent of E such that…”

“…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.…”

“…He proved that if Ω is bounded pseudoconvex domain of plurisubharmonic type m with C 2 boundary, φ ∈ C mα (∂Ω) with 0 < α ≤ 2 m and f 1 n ∈ C α (Ω) then M A(Ω, φ, f ) has a unique solution u ∈ C α (Ω). Guedj, Ko lodziej and Zeriahi [17] studied the problem in bounded strongly pseudoconvex domains. They showed that if φ ∈ C 1,1 (∂Ω) then the unique solution u to M A(Ω, φ, f ) is α-Hölder continuous on Ω, for any…”

“…Cuong [15] generalized the theorem of [17] to complex Hessian equation. Charabati [13] proved the Hölder regularity for solutions to M A(Ω, φ, f ) in bounded strongly hyperconvex Lipschitz domain.…”