Stability and strong convergence for complex Hessian equations with $L^1$ data

Abstract

We study complex $m$-Hessian equations on bounded hyperconvex domains with right-hand side in $L^1(Ω)$. The main contribution of this paper is a strong stability result for weak solutions in the Hessian energy sense. More precisely, if $f_j \to f$ in $L^1(Ω)$ and $u_j, u$ are the corresponding solutions, then \[ \int_Ω|u_j-u|\, H_m(u_j) \longrightarrow 0. \] This provides convergence in the natural energy topology associated to the Hessian operator, which is significantly stronger than convergence in capacity. For completeness, we also recall the existence of solutions and stability in capacity, which follow from known results in the literature.