On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds
Let \(X\) be a compact Kähler manifold of dimension \(n\), equipped with a big form \(\omega\) (a smooth semi-positive \((1,1)\)-form such that \(\int_X\omega^n>0\)). An upper semicontinuous function \(u\) is called \(\omega\)-psh if \(\omega_u\geq 0\), where \(\omega_u=\omega + i\partial\bar\partial u\). A stability result is obtained for \(\omega\)-psh solutions \(u\) of the equation \(\omega_u^n=f\omega^n\) for \(f\in L^p\) with respect to a regular volume form \(\Omega>0\), which generalizes a result by \textit{S. Kolodziej} [Indiana Univ. Math. J. 52, No.~3, 667--686 (2003; Zbl 1039.32050)]. In particular, an optimal stability exponent is found. In addition, a detailed proof is given for a result by \textit{Z. Zhang} [Int. Math. Res. Not. 2006, No.~11, Article ID 63640 (2006; Zbl 1112.32021)] on continuity of the solution to \(\omega_u^n=f\Omega\) with a special big form \(\omega\) and \(f\in L^p(\Omega)\).
scientific article; zbMATH DE number 5768610
scientific article