Viscosity solutions to complex Hessian equations

Abstract

We study viscosity solutions to complex hessian equations. In the local case, we consider $\Omega$ a bounded domain in ${\mathbb{C}}^n,$ $\beta$ the standard Kähler form in ${\mathcal{C}}^n$ and $1\le m\le n.$ Under some suitable conditions on $F,g$, we prove that the equation $(dd^c φ)^m\wedgeβ^{n-m}=F(x,φ)β^n,\ \f=g$ on $\pO$ admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum are Hölder continuous then so is the solution. In the global case, let $(X,\omega )$ be a compact hermitian homogeneous manifold where $\omega$ is an invariant hermitian metric (not necessarily Kähler). We prove that the equation $(\omega +d{d}^c\phi {)}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$ has a unique viscosity solution under some natural conditions on $F.$

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References in corpus (2)

• A Viscosity Approach to the Dirichlet Problem for Complex Monge-Ampère Equations
• Capacities and Hessians in a class of m-subharmonic functions