Viscosity solutions to complex Hessian equations

Abstract

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.