On the regularity of Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds

TL;DR

This paper establishes solvability and regularity results for the Dirichlet problem of fully non-linear elliptic equations on Hermitian manifolds, including boundary estimates and subsolution construction.

Contribution

It provides new boundary estimates and constructs subsolutions for complex non-linear elliptic equations on Hermitian manifolds, advancing the understanding of their regularity and solvability.

Findings

Established boundary estimates under subsolution assumptions

Proved solvability of the Dirichlet problem on Hermitian manifolds

Constructed explicit subsolutions on product manifolds

Abstract

We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.