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 $F.$

fix typos

References in corpus (5)

• Hölder continuous solutions to Monge-Ampère equations
• Viscosity solutions to degenerate Complex Monge-Ampère equations
• Solutions to degenerate complex Hessian equations
• A Viscosity Approach to the Dirichlet Problem for Complex Monge-Ampère Equations
• Capacities and Hessians in a class of m-subharmonic functions

Cited by in corpus (13)

• The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
• $C^{2,\alpha }$ estimates for nonlinear elliptic equations in complex and almost complex geometry
• On the singularity type of full mass currents in big cohomology classes
• Hölder continuous solutions to complex Hessian equations
• A viscosity approach to the Dirichlet problem for degenerate complex Hessian type equations
• A variational Approach to complex Hessian equations in ${\mathbb{C}}^n$
• The smoothing of $m$-subharmonic functions
• Viscosity solutions to quaternionic Monge-Ampère equations
• The Hölder continuous subsolution theorem for complex Hessian equations
• Viscosity solutions to parabolic complex Monge-Ampère equations
• Complex Hessian measures with respect to a background Hermitian form
• The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary
• On the viscosity approach to a class of fully nonlinear elliptic equations