A variational Approach to complex Hessian equations in ${\mathbb{C}}^n$

Abstract

Let $\Omega$ be a $m$-hyperconvex domain of ${\mathbb{C}}^n$ and $\beta$ be the standard Kähler form in ${\mathbb{C}}^n$. We introduce finite energy classes of $m$-subharmonic functions of Cegrell type, ${\mathcal{E}}_m^p,p>0$ and ${\mathcal{F}}_m$. Using a variational method we show that the degenerate complex Hessian equation $(d{d}^c\phi {)}^m\wedge {\beta }^{n-m}=\mu$ has a unique solution in ${\mathcal{E}}_m^1$ if and only if every function in ${\mathcal{E}}_m^1$ is integrable with respect to $\mu$. If $\mu$ has finite total mass and does not charge $m$-polar sets, then the equation has a unique solution in ${\mathcal{F}}_m$.

References in corpus (2)

• Viscosity solutions to complex Hessian equations
• Hölder continuous solutions to complex Hessian equations

Cited by in corpus (1)

• Complex Hessian Operator associated to an $m$-positive closed current and weighted $m$-capacity