THE CONTINUOUS SUBSOLUTION PROBLEM FOR COMPLEX HESSIAN EQUATIONS

Le problème des sous-solutions continues pour les équations hessiennes complexes

Mohamad Charabati (1) , Ahmed Zeriahi (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Let $\Omega \Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure on $\Omega$. We study the complex Hessian equation $(dd^c u)^m \wedge \beta^{n - m} = \mu$ on $\Omega$. First we give a sufficient condition on the measure $\mu$ in terms of its domination by the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in the proofs is to establish a new capacity estimate showing that the $m$-Hessian measure of a continuous $m$-subharmonic function on $\Omega$ with zero boundary values is dominated by an explicit function of the $m$-Hessian capacity with respect to $\Omega$, involving the modulus of continuity of $\varphi$. Another important ingredient is a new weak stability estimate on the Hessian measure of a continuous $m$-subharmonic function.

Mots clés: Complex Monge-Amp\`ere equations; complex Hessian equations; Dirichlet problem; subsolution; capacity estimates.

Domaines: Variables complexes [math.CV]

Mohamad Charabati, Ahmed Zeriahi. THE CONTINUOUS SUBSOLUTION PROBLEM FOR COMPLEX HESSIAN EQUATIONS. 2020. ⟨hal-02901096v2⟩