A priori estimate for the complex Monge-Ampère equation
The aim of this paper is to give a new proof of well-known results in pluripotential theory: the uniform estimate, stability and the Hölder continuity for solutions to the complex Monge-Ampère equation. The proofs presented in the article are based on various PDE techniques developed in the recent authors' paper [J. Funct. Anal. 279, No. 12, Article ID 108765 (2020; Zbl 1458.32037)]. Let \(\Omega\subset \mathbb C^n\) be a bounded and smooth strictly pseudoconvex domain with smooth boundary. For given functions \(\varphi\in \mathcal C(\partial \Omega)\) and \(f\in L^p(\Omega)\), \(p>1\), let \(U(\varphi,f)\) be the unique solution to the following Dirichlet problem for the complex Monge-Ampère equation \[ \begin{aligned}&(dd^cU(\varphi,f))^n=fdV_{2n},\\ &\lim_{\Omega\ni z\to \zeta\in \partial \Omega}U(\varphi,f)(z)=\varphi(\zeta),\end{aligned} \] where \(dV_{2n}\) is the Lebesgue measure in \(\mathbb C^n\). The following theorems are proved in the paper. Theorem 1.2 (uniform estimate). Assume that \(\varphi\in \mathcal C^{\infty}(\partial \Omega)\) and \(U(\varphi,f)\in\mathcal C^{\infty}(\bar \Omega)\). Then for any \(\delta \in \left(0,\frac {p-1}{np}\right)\) there exists a constant \(C\) depending only on \(n,p,\delta\) and \(diam(\Omega)\) such that \[ \left|\inf_{\Omega}U(\varphi,f)\right|\leq \left|\inf_{\partial \Omega}\varphi\right|+C||f||_{L^p}^{\frac 1n}V_{2n}(\Omega)^{\delta}. \] Theorem 1.3 (stability). Assume that \(\varphi,\psi\in \mathcal C(\partial \Omega)\), \(f,g\in L^p(\Omega)\) and \(U(\varphi,f), U(\psi,g)\in\mathcal C^{\infty}(\bar \Omega)\). Then for any \(\delta \in \left(0,\frac {p-1}{np}\right)\) there exists a constant \(C\) depending only on \(||f||_{L^p}\), \(||g||_{L^p}\), \(n,p,\delta\) and \(\mathrm{diam}\,(\Omega)\) such that \[ ||U(\varphi,f)-U(\psi,g)||_{L^{\infty}}\leq C\left(||f-g||_{L^1}^{\frac {\delta}{n(1+\delta)}}+||\varphi-\psi||_{L^{\infty}}^{\frac {\delta}{1+\delta}}\right). \] Theorem 1.4 (Hölder continuity). Assume that \(\varphi\in \mathcal C^{2\alpha}(\partial \Omega)\). If \(\Delta U(\varphi,0)\) has finite mass in \(\Omega\), then \(U(\varphi,f)\in \mathcal C^{\beta}\) for any \(\beta<\min\left(\alpha, \frac {2(p-1)}{p(n+1)}\right)\).
scientific article; zbMATH DE number 7363089
scientific article