Viscosity solutions to complex Hessian equations

We study viscosity solutions to complex Hessian equations. In the local case, we consider Ω a bounded domain in Cn, β the standard Kähler form in Cn and 1 m n. Under some suitable conditions on F, g, we prove that the equation (ddcϕ)m ∧βn−m = F(x,ϕ)βn, ϕ = g on ∂Ω admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum is Hölder continuous then so is the solution. In the global case, let (X,ω) be a compact Hermitian homogeneous manifold where ω is an invariant Hermitian metric (not necessarily Kähler).We prove that the equation (ω+ddcϕ)m ∧ωn−m = F(x,ϕ)ωn has a unique viscosity solution under some natural conditions on F. © 2013 Elsevier Inc. All rights reserved.