The Dirichlet problem for complex k-Hessian equations on compact Hermitian manifolds with boundary is solved under the assumption that a smooth subsolution exists.
Caffarelli, Interior a priori estimates for solutions of fully nonlinea r equations , Ann
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The Dirichlet Problem for the $k$-Hessian Equation on a complex manifold
The Dirichlet problem for complex k-Hessian equations on compact Hermitian manifolds with boundary is solved under the assumption that a smooth subsolution exists.