Viscosity solutions to complex Hessian equations

2012/09/30 by Hoang Chinh Lu, Chinh H. Lu, Lu Hoang Chinh · 43 citations
Mathematics · #Applied mathematics #First-order partial differential equation #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hessian equation #Hessian matrix #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Partial differential equation #Physics #Thermodynamics #Viscosity #Viscosity solution #math.AP #math.CV #math.DG

paper · pdf · doi:10.1016/j.jfa.2013.01.001

published in Journal of Functional Analysis 264(6), 1355-1379 (Elsevier BV) · fix typos

openalex publication_date 2013/01/29 · arxiv created 2013/02/06 · arxiv updated 2013/02/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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 (dd^c φ)^m\wedgeβ^n-m=F(x,φ)βⁿ, \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,ω) be a compact hermitian homogeneous manifold where ω is an invariant hermitian metric (not necessarily Kähler). We prove that the equation (ω+dd^cφ)^m\wedgeω^n-m=F(x,φ)ωⁿ has a unique viscosity solution under some natural conditions on F.

Citations

• Complex parallisable manifolds
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Cited by

• Very weak solutions of quadratic Hessian equations
• The Critical LYZ equation in Kähler Geometry
• Polar sets for m-subharmonic functions on compact Hermitian manifolds
• The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary
• The smoothing of m-subharmonic functions
• Viscosity solutions to quaternionic Monge-Ampère equations
• On the viscosity approach to a class of fully nonlinear elliptic equations
• The Monge-Ampère equation for (𝑛-1)-plurisubharmonic functions on a compact Kähler manifold

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