Modulus of continuity of solutions to complex Hessian equations

Abstract

We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order m (1 ≤ m ≤ n) with a continuous right hand side and a continuous boundary data in a bounded strongly m-pseudoconvex domain Ω ⋐ C n . Moreover when the right hand side is in L p (Ω), for some p > n/m and the boundary value function is C 1,1 we prove that the solution is Hölder continuous.

Hölder regularity for solutions to complex Monge–Ampère equations

“…When f ∈ L p (Ω) for p ≥ 2, we are able to find a Hölder continuous barrier to the Dirichlet problem with more better Hölder exponent. The following theorem was proved in [Ch14] for the complex Hessian equation and it is enough here to put m = n for the complex Monge-Ampère equation. We note that…”

Regularity of solutions to the Dirichlet problem for Monge-Ampere equations

“…The first part of this corollary was proved in Theorem 1.2 in [BKPZ15] with the Hölder exponent min{2γ, α}γ and the second part was proved in [GKZ08] and [Ch15] (see also [N14,Ch14] for the complex Hessian equation).…”

The Continuous Subsolution Problem for Complex Hessian Equations

“…To this end, we need to construct two functions v and w satifying the requirement of the Lemma 2.4. Let w be the maximal m-subharmonic function on Ω with boundary values g. By [Ch16b], we have κ w (δ) ≤ κ g ( √ δ) and by the comparison principle we have u ≤ w on Ω.…”