The Dirichlet problem for a complex Hessian equation on compact   Hermitian manifolds with boundary
  We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal K\"ahler metrics, provided a subsolution. 
Dongwei Gu and Ngoc Cuong Nguyen
math.DG math.AP math.CV